<html><head><title>[Nilmat] 3 Examples</title></head>
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<h1>3 Examples</h1><p>
<P>
<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP003.htm#SECT001">Constructing some nilpotent matrix groups</a>
<li> <A HREF="CHAP003.htm#SECT002">Testing nilpotency and other functions</a>
<li> <A HREF="CHAP003.htm#SECT003">Using the library of primitive nilpotent groups</a>
</ol><p>
<p>
<a name = "I0"></a>
In this chapter we give some examples of computing with the Package
<font face="Gill Sans,Helvetica,Arial">Nilmat</font>.
<p>
<p>
<h2><a name="SECT001">3.1 Constructing some nilpotent matrix groups</a></h2>
<p><p>
<pre>
gap> g1 := MaximalAbsolutelyIrreducibleNilpotentMatGroup(52,3,3);
<matrix group with 7 generators>
</pre>
<p>
The group <code>g1</code> is a subgroup of <i>GL</i>(52,3<sup>3</sup>) generated by 7 matrices.
<p>
<pre>
gap> g2 := MaximalAbsolutelyIrreducibleNilpotentMatGroup(180,11,2);
<matrix group with 41 generators>
</pre>
<p>
The group <code>g2</code> is a subgroup of <i>GL</i>(180,11<sup>2</sup>) generated by 41 matrices.
<p>
<pre>
gap> MaximalAbsolutelyIrreducibleNilpotentMatGroup(210,2,10);
fail
</pre>
<p>
In this third example, absolutely irreducible nilpotent subgroups of
<i>GL</i>(210,2<sup>10</sup>) do not exist, because the degree of the matrices
and the field size are both even.
<p>
<pre>
gap> g3 := MonomialNilpotentMatGroup(450);
<matrix group with 24 generators>
</pre>
<p>
Here <code>g3</code> is a monomial nilpotent subgroup of <i>GL</i>(450,<b>Q</b>).
<p>
<pre>
gap> g4 := ReducibleNilpotentReducibleMatGroup(3,180,11,2);
<matrix group with 82 generators>
</pre>
<p>
Here <tt>g</tt><tt>4</tt> < <i>GL</i>(540,11<sup>2</sup>) is the Kronecker product of a
unipotent subgroup of <i>GL</i>(3,11<sup>2</sup>) and the group <code>g2</code>.
<p>
<pre>
gap> g5 := ReducibleNilpotentMatGroup(7,36);
<matrix group with 72 generators>
</pre>
<p>
Here <tt>g</tt><tt>5</tt> < <i>GL</i>(252, <b>Q</b>) is a reducible nilpotent group constructed
as the Kronecker product of a unipotent subgroup of <i>GL</i>(7,<b>Q</b>) with
<code>MonomialNilpotentMatGroup(36)</code>.
<p>
<p>
<h2><a name="SECT002">3.2 Testing nilpotency and other functions</a></h2>
<p><p>
We now illustrate use of the functions
<code>IsNilpotentMatGroup</code>,
<code>SylowSubgroupsOfNilpotentFFMatGroup</code>,
<code>IsFiniteNilpotentMatGroup</code>,
<code>SizeOfNilpotentMatGroup</code>, and
<code>IsCompletelyReducibleNilpotentMatGroup</code>.
<p>
<pre>
gap> IsNilpotentMatGroup(GL(200,Integers));
false
gap> IsNilpotentMatGroup(GL(150,11^3));
false
gap> g6 := MaximalAbsolutelyIrreducibleNilpotentMatGroup(127,2,7);
<matrix group with 3 generators>
gap> IsNilpotentMatGroup(g6);
true
gap> g7 := MonomialNilpotentMatGroup(350);
<matrix group with 6 generators>
gap> IsNilpotentMatGroup(g7);
true
gap> IsFiniteNilpotentMatGroup(g7);
true
gap> g8 := ReducibleNilpotentMatGroup(6,35);
<matrix group with 5 generators>
gap> IsNilpotentMatGroup(g8);
true
gap> IsFiniteNilpotentMatGroup(g8);
false
gap> g9 := ReducibleNilpotentMatGroup(2,36,5,2);
<matrix group with 21 generators>
gap> SylowSubgroupsOfNilpotentFFMatGroup(g9);
[ <matrix group with 5 generators>, <matrix group with 6 generators>,
<matrix group with 1 generators> ]
gap> IsCompletelyReducibleNilpotentMatGroup(g9);
false
gap> g10 := MaximalAbsolutelyIrreducibleNilpotentMatGroup(24,5,2);
<matrix group with 17 generators>
gap> SizeOfNilpotentMatGroup(g10);
173946175488
gap> IsCompletelyReducibleNilpotentMatGroup(g10);
true
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