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<a id="X83827EDB7D36C407" name="X83827EDB7D36C407"></a>
<div class="ChapSects"><a href="chap6.html#X83827EDB7D36C407">6 New GAP Objects and Utility Functions provided by the
AtlasRep Package</a>
<div class="ContSect"> <a href="chap6.html#X8121E9567A7137C9">6.1 Straight Line Decisions</a>

<div class="ContSSBlock">
 <a href="chap6.html#X8787E2EC7DB85A89">6.1-1 IsStraightLineDecision</a>
 <a href="chap6.html#X82AFAD9F7FA5CE8A">6.1-2 LinesOfStraightLineDecision</a>
 <a href="chap6.html#X7B1A43427BD97FDF">6.1-3 NrInputsOfStraightLineDecision</a>
 <a href="chap6.html#X82A3632782E45F35">6.1-4 ScanStraightLineDecision</a>
 <a href="chap6.html#X825C4E4180F3D989">6.1-5 StraightLineDecision</a>
 <a href="chap6.html#X7E7B328A84685480">6.1-6 ResultOfStraightLineDecision</a>
 <a href="chap6.html#X7C94ECAC8583CEAE">6.1-7 Semi-Presentations and Presentations</a>

 <a href="chap6.html#X7C13D08C7D55E20A">6.1-8 AsStraightLineDecision</a>
 <a href="chap6.html#X7EA613C57DDC67D5">6.1-9 StraightLineProgramFromStraightLineDecision</a>
</div></div>
<div class="ContSect"> <a href="chap6.html#X7BE856BC785A9E8F">6.2 Black Box Programs</a>

<div class="ContSSBlock">
 <a href="chap6.html#X87CAF2DE870D0E3B">6.2-1 IsBBoxProgram</a>
 <a href="chap6.html#X7EA20532868F9863">6.2-2 ScanBBoxProgram</a>
 <a href="chap6.html#X7D211A5D8602B330">6.2-3 RunBBoxProgram</a>
 <a href="chap6.html#X869BACFB80A3CC87">6.2-4 ResultOfBBoxProgram</a>
 <a href="chap6.html#X826ACFE887E0B6B8">6.2-5 AsBBoxProgram</a>
 <a href="chap6.html#X7D36DFA87C8B2C48">6.2-6 AsStraightLineProgram</a>
</div></div>
<div class="ContSect"> <a href="chap6.html#X87E1F08D80C9E069">6.3 Representations of Minimal Degree</a>

<div class="ContSSBlock">
 <a href="chap6.html#X7DC66D8282B2BB7F">6.3-1 MinimalRepresentationInfo</a>
 <a href="chap6.html#X7E1B76DC86A8C405">6.3-2 MinimalRepresentationInfoData</a>
 <a href="chap6.html#X79C4C9F683E919C9">6.3-3 SetMinimalRepresentationInfo</a>
 <a href="chap6.html#X7FC33DFF8481F8D1">6.3-4 Criteria Used to Compute Minimality Information</a>

</div></div>
<div class="ContSect"> <a href="chap6.html#X7E5CB1637F127B2E">6.4 A JSON Interface</a>

<div class="ContSSBlock">
 <a href="chap6.html#X7870BBB184574D18">6.4-1 Why JSON?</a>

 <a href="chap6.html#X87A307D284975AA9">6.4-2 AGR.JsonText</a>
 <a href="chap6.html#X79DF4DC67DCFE27B">6.4-3 AGR.GapObjectOfJsonText</a>
</div></div>
</div>

<h3>6 New GAP Objects and Utility Functions provided by the
AtlasRep Package</h3>

This chapter describes GAP objects and functions that are provided by the AtlasRep package but that might be of general interest.

The new objects are straight line decisions (see Section <a href="chap6.html#X8121E9567A7137C9">6.1</a>) and black box programs (see Section <a href="chap6.html#X7BE856BC785A9E8F">6.2</a>).

The new functions are concerned with representations of minimal degree, see Section <a href="chap6.html#X87E1F08D80C9E069">6.3</a>, and a JSON interface, see Section <a href="chap6.html#X7E5CB1637F127B2E">6.4</a>.

<a id="X8121E9567A7137C9" name="X8121E9567A7137C9"></a>

<h4>6.1 Straight Line Decisions</h4>

Straight line decisions are similar to straight line programs (see Section <a href="../../../doc/ref/chap37.html#X7DC99E4284093FBB">Reference: Straight Line Programs</a>) but return <code class="keyw">true</code> or <code class="keyw">false</code>. A straight line decision checks whether its inputs have some property. An important example is to check whether a given list of group generators is in fact a list of standard generators (cf. Section<a href="chap3.html#X795DB7E486E0817D">3.3</a>) for this group.

A straight line decision in GAP is represented by an object in the filter <code class="func">IsStraightLineDecision</code> (<a href="chap6.html#X8787E2EC7DB85A89">6.1-1</a>) that stores a list of <q>lines</q> each of which has one of the following three forms.

<ol>
<li>a nonempty dense list l of integers,

</li>
<li>a pair [ l, i ] where l is a list of form 1. and i is a positive integer,

</li>
<li>a list [ <code class="code">"Order"</code>, i, n ] where i and n are positive integers.

</li>
</ol>
The first two forms have the same meaning as for straight line programs (see Section <a href="../../../doc/ref/chap37.html#X7DC99E4284093FBB">Reference: Straight Line Programs</a>), the last form means a check whether the element stored at the i-th label has the order n.

For the meaning of the list of lines, see <code class="func">ResultOfStraightLineDecision</code> (<a href="chap6.html#X7E7B328A84685480">6.1-6</a>).

Straight line decisions can be constructed using <code class="func">StraightLineDecision</code> (<a href="chap6.html#X825C4E4180F3D989">6.1-5</a>), defining attributes for straight line decisions are <code class="func">NrInputsOfStraightLineDecision</code> (<a href="chap6.html#X7B1A43427BD97FDF">6.1-3</a>) and <code class="func">LinesOfStraightLineDecision</code> (<a href="chap6.html#X82AFAD9F7FA5CE8A">6.1-2</a>), an operation for straight line decisions is <code class="func">ResultOfStraightLineDecision</code> (<a href="chap6.html#X7E7B328A84685480">6.1-6</a>).

Special methods applicable to straight line decisions are installed for the operations <code class="func">Display</code> (<a href="../../../doc/ref/chap6.html#X83A5C59278E13248">Reference: Display</a>), <code class="func">IsInternallyConsistent</code> (<a href="../../../doc/ref/chap12.html#X7F6C5C3287E8B816">Reference: IsInternallyConsistent</a>), <code class="func">PrintObj</code> (<a href="../../../doc/ref/chap6.html#X815BF22186FD43C9">Reference: PrintObj</a>), and <code class="func">ViewObj</code> (<a href="../../../doc/ref/chap6.html#X815BF22186FD43C9">Reference: ViewObj</a>).

For a straight line decision <var class="Arg">prog</var>, the default <code class="func">Display</code> (<a href="../../../doc/ref/chap6.html#X83A5C59278E13248">Reference: Display</a>) method prints the interpretation of <var class="Arg">prog</var> as a sequence of assignments of associative words and of order checks; a record with components <code class="code">gensnames</code> (with value a list of strings) and <code class="code">listname</code> (a string) may be entered as second argument of <code class="func">Display</code> (<a href="../../../doc/ref/chap6.html#X83A5C59278E13248">Reference: Display</a>), in this case these names are used, the default for <code class="code">gensnames</code> is <code class="code">[ g1, g2, </code>...<code class="code"> ]</code>, the default for <var class="Arg">listname</var> is r.

<a id="X8787E2EC7DB85A89" name="X8787E2EC7DB85A89"></a>

<h5>6.1-1 IsStraightLineDecision</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsStraightLineDecision</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
Each straight line decision in GAP lies in the filter <code class="func">IsStraightLineDecision</code>.

<a id="X82AFAD9F7FA5CE8A" name="X82AFAD9F7FA5CE8A"></a>

<h5>6.1-2 LinesOfStraightLineDecision</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LinesOfStraightLineDecision</code>( <var class="Arg">prog</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: the list of lines that define the straight line decision.

This defining attribute for the straight line decision <var class="Arg">prog</var> (see <code class="func">IsStraightLineDecision</code> (<a href="chap6.html#X8787E2EC7DB85A89">6.1-1</a>)) corresponds to <code class="func">LinesOfStraightLineProgram</code> (<a href="../../../doc/ref/chap37.html#X81A8AFC47F8E4B91">Reference: LinesOfStraightLineProgram</a>) for straight line programs.

<div class="example"><pre>
gap> dec:= StraightLineDecision( [ [ [ 1, 1, 2, 1 ], 3 ],
> [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ "Order", 3, 5 ] ] );
<straight line decision>
gap> LinesOfStraightLineDecision( dec );
[ [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 1, 2 ], [ "Order", 2, 3 ],
 [ "Order", 3, 5 ] ]
</pre></div>

<a id="X7B1A43427BD97FDF" name="X7B1A43427BD97FDF"></a>

<h5>6.1-3 NrInputsOfStraightLineDecision</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrInputsOfStraightLineDecision</code>( <var class="Arg">prog</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: the number of inputs required for the straight line decision.

This defining attribute corresponds to <code class="func">NrInputsOfStraightLineProgram</code> (<a href="../../../doc/ref/chap37.html#X820A592881D57802">Reference: NrInputsOfStraightLineProgram</a>).

<div class="example"><pre>
gap> NrInputsOfStraightLineDecision( dec );
2
</pre></div>

<a id="X82A3632782E45F35" name="X82A3632782E45F35"></a>

<h5>6.1-4 ScanStraightLineDecision</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ScanStraightLineDecision</code>( <var class="Arg">string</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: a record containing the straight line decision, or <code class="keyw">fail</code>.

Let <var class="Arg">string</var> be a string that encodes a straight line decision in the sense that it consists of the lines listed for <code class="func">ScanStraightLineProgram</code> (<a href="chap7.html#X7D6617E47B013A37">7.4-1</a>), except that <code class="code">oup</code> lines are not allowed, and instead lines of the following form may occur.

<dl>
<dt><code class="code">chor </code>a b</dt>
<dd>means that it is checked whether the order of the element at label a is b.

</dd>
</dl>
<code class="func">ScanStraightLineDecision</code> returns a record containing as the value of its component <code class="code">program</code> the corresponding GAP straight line decision (see <code class="func">IsStraightLineDecision</code> (<a href="chap6.html#X8787E2EC7DB85A89">6.1-1</a>)) if the input string satisfies the syntax rules stated above, and returns <code class="keyw">fail</code> otherwise. In the latter case, information about the first corrupted line of the program is printed if the info level of <code class="func">InfoCMeatAxe</code> (<a href="chap7.html#X78601C3A87921E08">7.1-2</a>) is at least 1.

<div class="example"><pre>
gap> str:= "inp 2\nchor 1 2\nchor 2 3\nmu 1 2 3\nchor 3 5";;
gap> prg:= ScanStraightLineDecision( str );
rec( program := <straight line decision> )
gap> prg:= prg.program;;
gap> Display( prg );
# input:
r:= [ g1, g2 ];
# program:
if Order( r[1] ) <> 2 then return false; fi;
if Order( r[2] ) <> 3 then return false; fi;
r[3]:= r[1]*r[2];
if Order( r[3] ) <> 5 then return false; fi;
# return value:
true
</pre></div>

<a id="X825C4E4180F3D989" name="X825C4E4180F3D989"></a>

<h5>6.1-5 StraightLineDecision</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StraightLineDecision</code>( <var class="Arg">lines</var>[, <var class="Arg">nrgens</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StraightLineDecisionNC</code>( <var class="Arg">lines</var>[, <var class="Arg">nrgens</var>] )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the straight line decision given by the list of lines.

Let <var class="Arg">lines</var> be a list of lists that defines a unique straight line decision (see <code class="func">IsStraightLineDecision</code> (<a href="chap6.html#X8787E2EC7DB85A89">6.1-1</a>)); in this case <code class="func">StraightLineDecision</code> returns this program, otherwise an error is signalled. The optional argument <var class="Arg">nrgens</var> specifies the number of input generators of the program; if a list of integers (a line of form 1. in the definition above) occurs in <var class="Arg">lines</var> then this number is not determined by <var class="Arg">lines</var> and therefore must be specified by the argument <var class="Arg">nrgens</var>; if not then <code class="func">StraightLineDecision</code> returns <code class="keyw">fail</code>.

<code class="func">StraightLineDecisionNC</code> does the same as <code class="func">StraightLineDecision</code>, except that the internal consistency of the program is not checked.

<a id="X7E7B328A84685480" name="X7E7B328A84685480"></a>

<h5>6.1-6 ResultOfStraightLineDecision</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResultOfStraightLineDecision</code>( <var class="Arg">prog</var>, <var class="Arg">gens</var>[, <var class="Arg">orderfunc</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: <code class="keyw">true</code> if all checks succeed, otherwise <code class="keyw">false</code>.

<code class="func">ResultOfStraightLineDecision</code> evaluates the straight line decision (see <code class="func">IsStraightLineDecision</code> (<a href="chap6.html#X8787E2EC7DB85A89">6.1-1</a>)) <var class="Arg">prog</var> at the group elements in the list <var class="Arg">gens</var>.

The function for computing the order of a group element can be given as the optional argument <var class="Arg">orderfunc</var>. For example, this may be a function that gives up at a certain limit if one has to be aware of extremely huge orders in failure cases.

The result of a straight line decision with lines p_1, p_2, ..., p_k when applied to <var class="Arg">gens</var> is defined as follows.

<dl>
<dt>(a)</dt>
<dd>First a list r of intermediate values is initialized with a shallow copy of <var class="Arg">gens</var>.

</dd>
<dt>(b)</dt>
<dd>For i ≤ k, before the i-th step, let r be of length n. If p_i is the external representation of an associative word in the first n generators then the image of this word under the homomorphism that is given by mapping r to these first n generators is added to r. If p_i is a pair [ l, j ], for a list l, then the same element is computed, but instead of being added to r, it replaces the j-th entry of r. If p_i is a triple [<code class="code">"Order"</code>, i, n ] then it is checked whether the order of r[i] is n; if not then <code class="keyw">false</code> is returned immediately.

</dd>
<dt>(c)</dt>
<dd>If all k lines have been processed and no order check has failed then <code class="keyw">true</code> is returned.

</dd>
</dl>
Here are some examples.

<div class="example"><pre>
gap> dec:= StraightLineDecision( [ ], 1 );
<straight line decision>
gap> ResultOfStraightLineDecision( dec, [ () ] );
true
</pre></div>

The above straight line decision <code class="code">dec</code> returns <code class="keyw">true</code> –for any input of the right length.

<div class="example"><pre>
gap> dec:= StraightLineDecision( [ [ [ 1, 1, 2, 1 ], 3 ],
> [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ "Order", 3, 5 ] ] );
<straight line decision>
gap> LinesOfStraightLineDecision( dec );
[ [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 1, 2 ], [ "Order", 2, 3 ],
 [ "Order", 3, 5 ] ]
gap> ResultOfStraightLineDecision( dec, [ (), () ] );
false
gap> ResultOfStraightLineDecision( dec, [ (1,2)(3,4), (1,4,5) ] );
true
</pre></div>

The above straight line decision admits two inputs; it tests whether the orders of the inputs are 2 and 3, and the order of their product is 5.

<a id="X7C94ECAC8583CEAE" name="X7C94ECAC8583CEAE"></a>

<h5>6.1-7 Semi-Presentations and Presentations</h5>

We can associate a finitely presented group F / R to each straight line decision <var class="Arg">dec</var>, say, as follows. The free generators of the free group F are in bijection with the inputs, and the defining relators generating R as a normal subgroup of F are given by those words w^k for which <var class="Arg">dec</var> contains a check whether the order of w equals k.

So if <var class="Arg">dec</var> returns <code class="keyw">true</code> for the input list [ g_1, g_2, ..., g_n ] then mapping the free generators of F to the inputs defines an epimorphism Φ from F to the group G, say, that is generated by these inputs, such that R is contained in the kernel of Φ.

(Note that <q>satisfying <var class="Arg">dec</var></q> is a stronger property than <q>satisfying a presentation</q>. For example, ⟨ x ∣ x^2 = x^3 = 1 ⟩ is a presentation for the trivial group, but the straight line decision that checks whether the order of x is both 2 and 3 clearly always returns <code class="keyw">false</code>.)

AtlasRep supports the following two kinds of straight line decisions.

<ul>
<li>A presentation is a straight line decision <var class="Arg">dec</var> that is defined for a set of standard generators of a group G and that returns <code class="keyw">true</code> if and only if the list of inputs is in fact a sequence of such standard generators for G. In other words, the relators derived from the order checks in the way described above are defining relators for G, and moreover these relators are words in terms of standard generators. (In particular the kernel of the map Φ equals R whenever <var class="Arg">dec</var> returns <code class="keyw">true</code>.)

</li>
<li>A semi-presentation is a straight line decision <var class="Arg">dec</var> that is defined for a set of standard generators of a group G and that returns <code class="keyw">true</code> for a list of inputs that is known to generate a group isomorphic with G if and only if these inputs form in fact a sequence of standard generators for G. In other words, the relators derived from the order checks in the way described above are not necessarily defining relators for G, but if we assume that the g_i generate G then they are standard generators. (In particular, F / R may be a larger group than G but in this case Φ maps the free generators of F to standard generators of G.)

More about semi-presentations can be found in <a href="chapBib.html#biBNW05">[NW05]</a>.

</li>
</ul>
Available presentations and semi-presentations are listed by <code class="func">DisplayAtlasInfo</code> (<a href="chap3.html#X79DACFFA7E2D1A99">3.5-1</a>), they can be accessed via <code class="func">AtlasProgram</code> (<a href="chap3.html#X801F2E657C8A79ED">3.5-4</a>). (Clearly each presentation is also a semi-presentation. So a semi-presentation for some standard generators of a group is regarded as available whenever a presentation for these standard generators and this group is available.)

Note that different groups can have the same semi-presentation. We illustrate this with an example that is mentioned in <a href="chapBib.html#biBNW05">[NW05]</a>. The groups L_2(7) ≅ L_3(2) and L_2(8) are generated by elements of the orders 2 and 3 such that their product has order 7, and no further conditions are necessary to define standard generators.

<div class="example"><pre>
gap> check:= AtlasProgram( "L2(8)", "check" );
rec( groupname := "L2(8)",
 identifier := [ "L2(8)", "L28G1-check1", 1, 1 ],
 program := <straight line decision>, standardization := 1,
 version := "1" )
gap> gens:= AtlasGenerators( "L2(8)", 1 );
rec( charactername := "1a+8a", constituents := [ 1, 6 ],
 contents := "core",
 generators := [ (1,2)(3,4)(6,7)(8,9), (1,3,2)(4,5,6)(7,8,9) ],
 groupname := "L2(8)", id := "",
 identifier := [ "L2(8)", [ "L28G1-p9B0.m1", "L28G1-p9B0.m2" ], 1, 9
 ], isPrimitive := true, maxnr := 1, p := 9, rankAction := 2,
 repname := "L28G1-p9B0", repnr := 1, size := 504,
 stabilizer := "2^3:7", standardization := 1, transitivity := 3,
 type := "perm" )
gap> ResultOfStraightLineDecision( check.program, gens.generators );
true
gap> gens:= AtlasGenerators( "L3(2)", 1 );
rec( contents := "core", generators := [ (2,4)(3,5), (1,2,3)(5,6,7) ],
 groupname := "L3(2)", id := "a",
 identifier := [ "L3(2)", [ "L27G1-p7aB0.m1", "L27G1-p7aB0.m2" ], 1,
 7 ], isPrimitive := true, maxnr := 1, p := 7, rankAction := 2,
 repname := "L27G1-p7aB0", repnr := 1, size := 168,
 stabilizer := "S4", standardization := 1, transitivity := 2,
 type := "perm" )
gap> ResultOfStraightLineDecision( check.program, gens.generators );
true
</pre></div>

<a id="X7C13D08C7D55E20A" name="X7C13D08C7D55E20A"></a>

<h5>6.1-8 AsStraightLineDecision</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsStraightLineDecision</code>( <var class="Arg">bbox</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: an equivalent straight line decision for the given black box program, or <code class="keyw">fail</code>.

For a black box program (see <code class="func">IsBBoxProgram</code> (<a href="chap6.html#X87CAF2DE870D0E3B">6.2-1</a>)) <var class="Arg">bbox</var>, <code class="func">AsStraightLineDecision</code> returns a straight line decision (see <code class="func">IsStraightLineDecision</code> (<a href="chap6.html#X8787E2EC7DB85A89">6.1-1</a>)) with the same output as <var class="Arg">bbox</var>, in the sense of <code class="func">AsBBoxProgram</code> (<a href="chap6.html#X826ACFE887E0B6B8">6.2-5</a>), if such a straight line decision exists, and <code class="keyw">fail</code> otherwise.

<div class="example"><pre>
gap> lines:= [ [ "Order", 1, 2 ], [ "Order", 2, 3 ],
> [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 3, 5 ] ];;
gap> dec:= StraightLineDecision( lines, 2 );
<straight line decision>
gap> bboxdec:= AsBBoxProgram( dec );
<black box program>
gap> asdec:= AsStraightLineDecision( bboxdec );
<straight line decision>
gap> LinesOfStraightLineDecision( asdec );
[ [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ [ 1, 1, 2, 1 ], 3 ],
 [ "Order", 3, 5 ] ]
</pre></div>

<a id="X7EA613C57DDC67D5" name="X7EA613C57DDC67D5"></a>

<h5>6.1-9 StraightLineProgramFromStraightLineDecision</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StraightLineProgramFromStraightLineDecision</code>( <var class="Arg">dec</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: the straight line program associated to the given straight line decision.

For a straight line decision <var class="Arg">dec</var> (see <code class="func">IsStraightLineDecision</code> (<a href="chap6.html#X8787E2EC7DB85A89">6.1-1</a>), <code class="func">StraightLineProgramFromStraightLineDecision</code> returns the straight line program (see <code class="func">IsStraightLineProgram</code> (<a href="../../../doc/ref/chap37.html#X7F69FF3F7C6694CB">Reference: IsStraightLineProgram</a>) obtained by replacing each line of type 3. (i.e, each order check) by an assignment of the power in question to a new slot, and by declaring the list of these elements as the return value.

This means that the return value describes exactly the defining relators of the presentation that is associated to the straight line decision, see <a href="chap6.html#X7C94ECAC8583CEAE">6.1-7</a>.

For example, one can use the return value for printing the relators with <code class="func">StringOfResultOfStraightLineProgram</code> (<a href="../../../doc/ref/chap37.html#X8098EAAF7D344466">Reference: StringOfResultOfStraightLineProgram</a>), or for explicitly constructing the relators as words in terms of free generators, by applying <code class="func">ResultOfStraightLineProgram</code> (<a href="../../../doc/ref/chap37.html#X7847D32B863E822F">Reference: ResultOfStraightLineProgram</a>) to the program and to these generators.

<div class="example"><pre>
gap> dec:= StraightLineDecision( [ [ [ 1, 1, 2, 1 ], 3 ],
> [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ "Order", 3, 5 ] ] );
<straight line decision>
gap> prog:= StraightLineProgramFromStraightLineDecision( dec );
<straight line program>
gap> Display( prog );
# input:
r:= [ g1, g2 ];
# program:
r[3]:= r[1]*r[2];
r[4]:= r[1]^2;
r[5]:= r[2]^3;
r[6]:= r[3]^5;
# return values:
[ r[4], r[5], r[6] ]
gap> StringOfResultOfStraightLineProgram( prog, [ "a", "b" ] );
"[ a^2, b^3, (ab)^5 ]"
gap> gens:= GeneratorsOfGroup( FreeGroup( "a", "b" ) );
[ a, b ]
gap> ResultOfStraightLineProgram( prog, gens );
[ a^2, b^3, (a*b)^5 ]
</pre></div>

<a id="X7BE856BC785A9E8F" name="X7BE856BC785A9E8F"></a>

<h4>6.2 Black Box Programs</h4>

Black box programs formalize the idea that one takes some group elements, forms arithmetic expressions in terms of them, tests properties of these expressions, executes conditional statements (including jumps inside the program) depending on the results of these tests, and eventually returns some result.

A specification of the language can be found in <a href="chapBib.html#biBNic06">[Nic06]</a>, see also

<a href="http://atlas.math.rwth-aachen.de/Atlas/info/blackbox.html">http://atlas.math.rwth-aachen.de/Atlas/info/blackbox.html</a>.

The inputs of a black box program may be explicit group elements, and the program may also ask for random elements from a given group. The program steps form products, inverses, conjugates, commutators, etc. of known elements, tests concern essentially the orders of elements, and the result is a list of group elements or <code class="keyw">true</code> or <code class="keyw">false</code> or <code class="keyw">fail</code>.

Examples that can be modeled by black box programs are

<dl>
<dt>straight line programs,</dt>
<dd>which require a fixed number of input elements and form arithmetic expressions of elements but do not use random elements, tests, conditional statements and jumps; the return value is always a list of elements; these programs are described in Section <a href="../../../doc/ref/chap37.html#X7DC99E4284093FBB">Reference: Straight Line Programs</a>.

</dd>
<dt>straight line decisions,</dt>
<dd>which differ from straight line programs only in the sense that also order tests are admissible, and that the return value is <code class="keyw">true</code> if all these tests are satisfied, and <code class="keyw">false</code> as soon as the first such test fails; they are described in Section <a href="chap6.html#X8121E9567A7137C9">6.1</a>.

</dd>
<dt>scripts for finding standard generators,</dt>
<dd>which take a group and a function to generate a random element in this group but no explicit input elements, admit all control structures, and return either a list of standard generators or <code class="keyw">fail</code>; see <code class="func">ResultOfBBoxProgram</code> (<a href="chap6.html#X869BACFB80A3CC87">6.2-4</a>) for examples.

</dd>
</dl>
In the case of general black box programs, currently GAP provides only the possibility to read an existing program via <code class="func">ScanBBoxProgram</code> (<a href="chap6.html#X7EA20532868F9863">6.2-2</a>), and to run the program using <code class="func">RunBBoxProgram</code> (<a href="chap6.html#X7D211A5D8602B330">6.2-3</a>). It is not our aim to write such programs in GAP.

The special case of the <q>find</q> scripts mentioned above is also admissible as an argument of <code class="func">ResultOfBBoxProgram</code> (<a href="chap6.html#X869BACFB80A3CC87">6.2-4</a>), which returns either the set of found generators or <code class="keyw">fail</code>.

Contrary to the general situation, more support is provided for straight line programs and straight line decisions in GAP, see Section <a href="../../../doc/ref/chap37.html#X7DC99E4284093FBB">Reference: Straight Line Programs</a> for functions that manipulate them (compose, restrict etc.).

The functions <code class="func">AsStraightLineProgram</code> (<a href="chap6.html#X7D36DFA87C8B2C48">6.2-6</a>) and <code class="func">AsStraightLineDecision</code> (<a href="chap6.html#X7C13D08C7D55E20A">6.1-8</a>) can be used to transform a general black box program object into a straight line program or a straight line decision if this is possible.

Conversely, one can create an equivalent general black box program from a straight line program or from a straight line decision with <code class="func">AsBBoxProgram</code> (<a href="chap6.html#X826ACFE887E0B6B8">6.2-5</a>).

Computing a straight line program related to a given straight line decision is supported in the sense of <code class="func">StraightLineProgramFromStraightLineDecision</code> (<a href="chap6.html#X7EA613C57DDC67D5">6.1-9</a>).

Note that none of these three kinds of objects is a special case of another: Running a black box program with <code class="func">RunBBoxProgram</code> (<a href="chap6.html#X7D211A5D8602B330">6.2-3</a>) yields a record, running a straight line program with <code class="func">ResultOfStraightLineProgram</code> (<a href="../../../doc/ref/chap37.html#X7847D32B863E822F">Reference: ResultOfStraightLineProgram</a>) yields a list of elements, and running a straight line decision with <code class="func">ResultOfStraightLineDecision</code> (<a href="chap6.html#X7E7B328A84685480">6.1-6</a>) yields <code class="keyw">true</code> or <code class="keyw">false</code>.

<a id="X87CAF2DE870D0E3B" name="X87CAF2DE870D0E3B"></a>

<h5>6.2-1 IsBBoxProgram</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsBBoxProgram</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
Each black box program in GAP lies in the filter <code class="func">IsBBoxProgram</code>.

<a id="X7EA20532868F9863" name="X7EA20532868F9863"></a>

<h5>6.2-2 ScanBBoxProgram</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ScanBBoxProgram</code>( <var class="Arg">string</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: a record containing the black box program encoded by the input string, or <code class="keyw">fail</code>.

For a string <var class="Arg">string</var> that describes a black box program, e.g., the return value of <code class="func">StringFile</code> (<a href="../../../pkg/gapdoc/doc/chap6.html#X7E14D32181FBC3C3">GAPDoc: StringFile</a>), <code class="func">ScanBBoxProgram</code> computes this black box program. If this is successful then the return value is a record containing as the value of its component <code class="code">program</code> the corresponding GAP object that represents the program, otherwise <code class="keyw">fail</code> is returned.

As the first example, we construct a black box program that tries to find standard generators for the alternating group A_5; these standard generators are any pair of elements of the orders 2 and 3, respectively, such that their product has order 5.

<div class="example"><pre>
gap> findstr:= "\
> set V 0\n\
> lbl START1\n\
> rand 1\n\
> ord 1 A\n\
> incr V\n\
> if V gt 100 then timeout\n\
> if A notin 1 2 3 5 then fail\n\
> if A noteq 2 then jmp START1\n\
> lbl START2\n\
> rand 2\n\
> ord 2 B\n\
> incr V\n\
> if V gt 100 then timeout\n\
> if B notin 1 2 3 5 then fail\n\
> if B noteq 3 then jmp START2\n\
> # The elements 1 and 2 have the orders 2 and 3, respectively.\n\
> set X 0\n\
> lbl CONJ\n\
> incr X\n\
> if X gt 100 then timeout\n\
> rand 3\n\
> cjr 2 3\n\
> mu 1 2 4 # ab\n\
> ord 4 C\n\
> if C notin 2 3 5 then fail\n\
> if C noteq 5 then jmp CONJ\n\
> oup 2 1 2";;
gap> find:= ScanBBoxProgram( findstr );
rec( program := <black box program> )
</pre></div>

The second example is a black box program that checks whether its two inputs are standard generators for A_5.

<div class="example"><pre>
gap> checkstr:= "\
> chor 1 2\n\
> chor 2 3\n\
> mu 1 2 3\n\
> chor 3 5";;
gap> check:= ScanBBoxProgram( checkstr );
rec( program := <black box program> )
</pre></div>

<a id="X7D211A5D8602B330" name="X7D211A5D8602B330"></a>

<h5>6.2-3 RunBBoxProgram</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RunBBoxProgram</code>( <var class="Arg">prog</var>, <var class="Arg">G</var>, <var class="Arg">input</var>, <var class="Arg">options</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: a record describing the result and the statistics of running the black box program <var class="Arg">prog</var>, or <code class="keyw">fail</code>, or the string <code class="code">"timeout"</code>.

For a black box program <var class="Arg">prog</var>, a group <var class="Arg">G</var>, a list <var class="Arg">input</var> of group elements, and a record <var class="Arg">options</var>, <code class="func">RunBBoxProgram</code> applies <var class="Arg">prog</var> to <var class="Arg">input</var>, where <var class="Arg">G</var> is used only to compute random elements.

The return value is <code class="keyw">fail</code> if a syntax error or an explicit <code class="code">fail</code> statement is reached at runtime, and the string <code class="code">"timeout"</code> if a <code class="code">timeout</code> statement is reached. (The latter might mean that the random choices were unlucky.) Otherwise a record with the following components is returned.

<dl>
<dt><code class="code">gens</code></dt>
<dd>a list of group elements, bound if an <code class="code">oup</code> statement was reached,

</dd>
<dt><code class="code">result</code></dt>
<dd><code class="keyw">true</code> if a <code class="code">true</code> statement was reached, <code class="keyw">false</code> if either a <code class="code">false</code> statement or a failed order check was reached,

</dd>
</dl>
The other components serve as statistical information about the numbers of the various operations (<code class="code">multiply</code>, <code class="code">invert</code>, <code class="code">power</code>, <code class="code">order</code>, <code class="code">random</code>, <code class="code">conjugate</code>, <code class="code">conjugateinplace</code>, <code class="code">commutator</code>), and the runtime in milliseconds (<code class="code">timetaken</code>).

The following components of <var class="Arg">options</var> are supported.

<dl>
<dt><code class="code">randomfunction</code></dt>
<dd>the function called with argument <var class="Arg">G</var> in order to compute a random element of <var class="Arg">G</var> (default <code class="func">PseudoRandom</code> (<a href="../../../doc/ref/chap30.html#X811B5BD47DC5356B">Reference: PseudoRandom</a>))

</dd>
<dt><code class="code">orderfunction</code></dt>
<dd>the function for computing element orders (default <code class="func">Order</code> (<a href="../../../doc/ref/chap31.html#X84F59A2687C62763">Reference: Order</a>)),

</dd>
<dt><code class="code">quiet</code></dt>
<dd>if <code class="keyw">true</code> then ignore <code class="code">echo</code> statements (default <code class="keyw">false</code>),

</dd>
<dt><code class="code">verbose</code></dt>
<dd>if <code class="keyw">true</code> then print information about the line that is currently processed, and about order checks (default <code class="keyw">false</code>),

</dd>
<dt><code class="code">allowbreaks</code></dt>
<dd>if <code class="keyw">true</code> then call <code class="func">Error</code> (<a href="../../../doc/ref/chap6.html#X7E7AD8D87EBA1A08">Reference: Error</a>) when a <code class="code">break</code> statement is reached, otherwise ignore <code class="code">break</code> statements (default <code class="keyw">true</code>).

</dd>
</dl>
As an example, we run the black box programs constructed in the example for <code class="func">ScanBBoxProgram</code> (<a href="chap6.html#X7EA20532868F9863">6.2-2</a>).

<div class="example"><pre>
gap> g:= AlternatingGroup( 5 );;
gap> res:= RunBBoxProgram( find.program, g, [], rec() );;
gap> IsBound( res.gens ); IsBound( res.result );
true
false
gap> List( res.gens, Order );
[ 2, 3 ]
gap> Order( Product( res.gens ) );
5
gap> res:= RunBBoxProgram( check.program, "dummy", res.gens, rec() );;
gap> IsBound( res.gens ); IsBound( res.result );
false
true
gap> res.result;
true
gap> othergens:= GeneratorsOfGroup( g );;
gap> res:= RunBBoxProgram( check.program, "dummy", othergens, rec() );;
gap> res.result;
false
</pre></div>

<a id="X869BACFB80A3CC87" name="X869BACFB80A3CC87"></a>

<h5>6.2-4 ResultOfBBoxProgram</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResultOfBBoxProgram</code>( <var class="Arg">prog</var>, <var class="Arg">G</var>[, <var class="Arg">options</var>] )</td><td class="tdright">( function )</td></tr></table></div>
Returns: a list of group elements or <code class="keyw">true</code>, <code class="keyw">false</code>, <code class="keyw">fail</code>, or the string <code class="code">"timeout"</code>.

This function calls <code class="func">RunBBoxProgram</code> (<a href="chap6.html#X7D211A5D8602B330">6.2-3</a>) with the black box program <var class="Arg">prog</var> and second argument either a group or a list of group elements; if <var class="Arg">options</var> is not given then the default options of <code class="func">RunBBoxProgram</code> (<a href="chap6.html#X7D211A5D8602B330">6.2-3</a>) are assumed. The return value is <code class="keyw">fail</code> if this call yields <code class="keyw">fail</code>, otherwise the <code class="code">gens</code> component of the result, if bound, or the <code class="code">result</code> component if not.

Note that a group <var class="Arg">G</var> is used as the second argument in the call of <code class="func">RunBBoxProgram</code> (<a href="chap6.html#X7D211A5D8602B330">6.2-3</a>) (the source for random elements), whereas a list <var class="Arg">G</var> is used as the third argument (the inputs).

As an example, we run the black box programs constructed in the example for <code class="func">ScanBBoxProgram</code> (<a href="chap6.html#X7EA20532868F9863">6.2-2</a>).

<div class="example"><pre>
gap> g:= AlternatingGroup( 5 );;
gap> res:= ResultOfBBoxProgram( find.program, g );;
gap> List( res, Order );
[ 2, 3 ]
gap> Order( Product( res ) );
5
gap> res:= ResultOfBBoxProgram( check.program, res );
true
gap> othergens:= GeneratorsOfGroup( g );;
gap> res:= ResultOfBBoxProgram( check.program, othergens );
false
</pre></div>

<a id="X826ACFE887E0B6B8" name="X826ACFE887E0B6B8"></a>

<h5>6.2-5 AsBBoxProgram</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsBBoxProgram</code>( <var class="Arg">slp</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: an equivalent black box program for the given straight line program or straight line decision.

Let <var class="Arg">slp</var> be a straight line program (see <code class="func">IsStraightLineProgram</code> (<a href="../../../doc/ref/chap37.html#X7F69FF3F7C6694CB">Reference: IsStraightLineProgram</a>)) or a straight line decision (see <code class="func">IsStraightLineDecision</code> (<a href="chap6.html#X8787E2EC7DB85A89">6.1-1</a>)). Then <code class="func">AsBBoxProgram</code> returns a black box program <var class="Arg">bbox</var> (see <code class="func">IsBBoxProgram</code> (<a href="chap6.html#X87CAF2DE870D0E3B">6.2-1</a>)) with the <q>same</q> output as <var class="Arg">slp</var>, in the sense that <code class="func">ResultOfBBoxProgram</code> (<a href="chap6.html#X869BACFB80A3CC87">6.2-4</a>) yields the same result for <var class="Arg">bbox</var> as <code class="func">ResultOfStraightLineProgram</code> (<a href="../../../doc/ref/chap37.html#X7847D32B863E822F">Reference: ResultOfStraightLineProgram</a>) or <code class="func">ResultOfStraightLineDecision</code> (<a href="chap6.html#X7E7B328A84685480">6.1-6</a>), respectively, for <var class="Arg">slp</var>.

<div class="example"><pre>
gap> f:= FreeGroup( "x", "y" );; gens:= GeneratorsOfGroup( f );;
gap> slp:= StraightLineProgram( [ [1,2,2,3], [3,-1] ], 2 );
<straight line program>
gap> ResultOfStraightLineProgram( slp, gens );
y^-3*x^-2
gap> bboxslp:= AsBBoxProgram( slp );
<black box program>
gap> ResultOfBBoxProgram( bboxslp, gens );
[ y^-3*x^-2 ]
gap> lines:= [ [ "Order", 1, 2 ], [ "Order", 2, 3 ],
> [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 3, 5 ] ];;
gap> dec:= StraightLineDecision( lines, 2 );
<straight line decision>
gap> ResultOfStraightLineDecision( dec, [ (1,2)(3,4), (1,3,5) ] );
true
gap> ResultOfStraightLineDecision( dec, [ (1,2)(3,4), (1,3,4) ] );
false
gap> bboxdec:= AsBBoxProgram( dec );
<black box program>
gap> ResultOfBBoxProgram( bboxdec, [ (1,2)(3,4), (1,3,5) ] );
true
gap> ResultOfBBoxProgram( bboxdec, [ (1,2)(3,4), (1,3,4) ] );
false
</pre></div>

<a id="X7D36DFA87C8B2C48" name="X7D36DFA87C8B2C48"></a>

<h5>6.2-6 AsStraightLineProgram</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsStraightLineProgram</code>( <var class="Arg">bbox</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: an equivalent straight line program for the given black box program, or <code class="keyw">fail</code>.

For a black box program (see <code class="func">AsBBoxProgram</code> (<a href="chap6.html#X826ACFE887E0B6B8">6.2-5</a>)) <var class="Arg">bbox</var>, <code class="func">AsStraightLineProgram</code> returns a straight line program (see <code class="func">IsStraightLineProgram</code> (<a href="../../../doc/ref/chap37.html#X7F69FF3F7C6694CB">Reference: IsStraightLineProgram</a>)) with the same output as <var class="Arg">bbox</var> if such a straight line program exists, and <code class="keyw">fail</code> otherwise.

<div class="example"><pre>
gap> Display( AsStraightLineProgram( bboxslp ) );
# input:
r:= [ g1, g2 ];
# program:
r[3]:= r[1]^2;
r[4]:= r[2]^3;
r[5]:= r[3]*r[4];
r[3]:= r[5]^-1;
# return values:
[ r[3] ]
gap> AsStraightLineProgram( bboxdec );
fail
</pre></div>

<a id="X87E1F08D80C9E069" name="X87E1F08D80C9E069"></a>

<h4>6.3 Representations of Minimal Degree</h4>

This section deals with minimal degrees of permutation and matrix representations. We do not provide an algorithm that computes these degrees for an arbitrary group, we only provide some tools for evaluating known databases, mainly concerning <q>bicyclic extensions</q> (see <a href="chapBib.html#biBCCN85">[CCN+85, Section 6.5]</a>) of simple groups, in order to derive the minimal degrees, see Section <a href="chap6.html#X7FC33DFF8481F8D1">6.3-4</a>.

In the AtlasRep package, this information can be used for prescribing <q>minimality conditions</q> in <code class="func">DisplayAtlasInfo</code> (<a href="chap3.html#X79DACFFA7E2D1A99">3.5-1</a>), <code class="func">OneAtlasGeneratingSetInfo</code> (<a href="chap3.html#X841478AB7CD06D44">3.5-6</a>), and <code class="func">AllAtlasGeneratingSetInfos</code> (<a href="chap3.html#X84C2D76482E60E42">3.5-7</a>). An overview of the stored minimal degrees can be shown with <code class="func">BrowseMinimalDegrees</code> (<a href="chap3.html#X7F31A7CB841FE63F">3.6-1</a>).

<a id="X7DC66D8282B2BB7F" name="X7DC66D8282B2BB7F"></a>

<h5>6.3-1 MinimalRepresentationInfo</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MinimalRepresentationInfo</code>( <var class="Arg">grpname</var>, <var class="Arg">conditions</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: a record with the components <code class="code">value</code> and <code class="code">source</code>, or <code class="keyw">fail</code>

Let <var class="Arg">grpname</var> be the GAP name of a group G, say. If the information described by <var class="Arg">conditions</var> about minimal representations of this group can be computed or is stored then <code class="func">MinimalRepresentationInfo</code> returns a record with the components <code class="code">value</code> and <code class="code">source</code>, otherwise <code class="keyw">fail</code> is returned.

The following values for <var class="Arg">conditions</var> are supported.

<ul>
<li>If <var class="Arg">conditions</var> is <code class="func">NrMovedPoints</code> (<a href="../../../doc/ref/chap42.html#X85E7B1E28430F49E">Reference: NrMovedPoints for a permutation</a>) then <code class="code">value</code>, if known, is the degree of a minimal faithful (not necessarily transitive) permutation representation for G.

</li>
<li>If <var class="Arg">conditions</var> consists of <code class="func">Characteristic</code> (<a href="../../../doc/ref/chap31.html#X81278E53800BF64D">Reference: Characteristic</a>) and a prime integer <var class="Arg">p</var> then <code class="code">value</code>, if known, is the dimension of a minimal faithful (not necessarily irreducible) matrix representation in characteristic <var class="Arg">p</var> for G.

</li>
<li>If <var class="Arg">conditions</var> consists of <code class="func">Size</code> (<a href="../../../doc/ref/chap30.html#X858ADA3B7A684421">Reference: Size</a>) and a prime power <var class="Arg">q</var> then <code class="code">value</code>, if known, is the dimension of a minimal faithful (not necessarily irreducible) matrix representation over the field of size <var class="Arg">q</var> for G.

</li>
</ul>
In all cases, the value of the component <code class="code">source</code> is a list of strings that describe sources of the information, which can be the ordinary or modular character table of G (see <a href="chapBib.html#biBCCN85">[CCN+85]</a>, <a href="chapBib.html#biBJLPW95">[JLPW95]</a>, <a href="chapBib.html#biBHL89">[HL89]</a>), the table of marks of G, or <a href="chapBib.html#biBJan05">[Jan05]</a>. For an overview of minimal degrees of faithful matrix representations for sporadic simple groups and their covering groups, see also

<a href="http://www.math.rwth-aachen.de/~MOC/mindeg/">http://www.math.rwth-aachen.de/~MOC/mindeg/</a>.

Note that <code class="func">MinimalRepresentationInfo</code> cannot provide any information about minimal representations over prescribed fields in characteristic zero.

Information about groups that occur in the AtlasRep package is precomputed in <code class="func">MinimalRepresentationInfoData</code> (<a href="chap6.html#X7E1B76DC86A8C405">6.3-2</a>), so the packages CTblLib and TomLib are not needed when <code class="func">MinimalRepresentationInfo</code> is called for these groups. (The only case that is not covered by this list is that one asks for the minimal degree of matrix representations over a prescribed field in characteristic coprime to the group order.)

One of the following strings can be given as an additional last argument.

<dl>
<dt><code class="code">"cache"</code></dt>
<dd>means that the function tries to compute (and then store) values that are not stored in <code class="func">MinimalRepresentationInfoData</code> (<a href="chap6.html#X7E1B76DC86A8C405">6.3-2</a>), but stored values are preferred; this is also the default.

</dd>
<dt><code class="code">"lookup"</code></dt>
<dd>means that stored values are returned but the function does not attempt to compute values that are not stored in <code class="func">MinimalRepresentationInfoData</code> (<a href="chap6.html#X7E1B76DC86A8C405">6.3-2</a>).

</dd>
<dt><code class="code">"recompute"</code></dt>
<dd>means that the function always tries to compute the desired value, and checks the result against stored values.

</dd>
</dl>

<div class="example"><pre>
gap> MinimalRepresentationInfo( "A5", NrMovedPoints );
rec(
 source := [ "computed (alternating group)",
 "computed (char. table)", "computed (subgroup tables)",
 "computed (subgroup tables, known repres.)",
 "computed (table of marks)" ], value := 5 )
gap> MinimalRepresentationInfo( "A5", Characteristic, 2 );
rec( source := [ "computed (char. table)" ], value := 2 )
gap> MinimalRepresentationInfo( "A5", Size, 2 );
--> --------------------

--> maximum size reached

--> --------------------

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