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href="aboutContents.html">Table
Of 
Contents</a> 
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 <td
style="text-align: center; vertical-align: top; font-weight: bold; color: rgb(0, 0, 102);"><big>About
HAP: Overview 
 </big></td>
 <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><a
href="aboutDefinitions.html">Next</a> 
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 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">HAP
can be used to make basic calculations in the cohomology of finite and
infinite groups.
For example, to calculate the integral homology Hn(D201,Z)
of the dihedral group of order 402 in dimension n=99 we could perform
the following commands. 
 </td>
 </tr>
 <tr>
 <td
style="width: 30%; background-color: rgb(255, 255, 204); vertical-align: top;">gap>
F:=FreeGroup(2);; x:=F.1;; y:=F.2;; 
 
gap>
G:=F/[x^2,y^201,(x*y)^2];; G:=Image(IsomorphismPermGroup(G));; 
 
gap> GroupHomology(G,99); 
[ 2, 3, 67 ] 
 
gap> time; 
4845 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
HAP command GroupHomology(G,n)
returns the abelian group
invariants of the n-dimensional homology of the group G with
coefficients in the integers Z with trivial G-action. We see that H99(D201,Z)
= Z402, (Timings are in milliseconds, and most are measured
on a
1.4GHz laptop with 256MB memory.) 
 
The above example has two features that dramatically help the
computations.
Firstly, D201 is a relatively small group. Secondly, D201
has periodic homology with period 4 (meaning that Hn(D201,Z)
= Hn+4(D201,Z) for n>0)
and so the homology groups themselves are small. 
 
Typically, the homology of larger non-periodic groups can only
be computed in low dimensions. The following commands show that: 
 <ul>
 <li>the
alternating group A7 (of order 2520) has H10(A7,Z)
= Z6+(Z3)2 ,</li>
 </ul>
 <ul>
 <li>the special linear group SL3(Z3) (of
order 5616) has H8(SL3(Z3),Z) = Z6
,</li>
 </ul>
 <ul>
 <li>the Coxeter group B5 (of order 3840),
represented by a Coxeter diagram on 5 vertices, has H4(B5,Z)
= (Z2)12
.</li>
 <li>the group K=Ker( SL2(Z53)
→ SL2(Z5) ) (of order 15625) has H3(K,Z)
= (Z5)6+Z125.
(This reproduces a calculation of W.Browder and J.Pakianathan which was
used to produce a <a href="aboutExtensions.html#Adem">counter-example</a>
to a conjecture of A. Adem.) 
 </li>
 <li>the abelian group G=C2×C4×C6×C8×C10
 ×C12 (of order 46080) has H6(G,Z)
= (Z2)280+(Z4)12+(Z12)3
. 
 </li>
 <li>the Mathieu simple group M23 (of order 10200960)
has H2(M23,Z) = H3(M23,Z) =
H4(M23,Z) =
0; this reproduces J. Milgram's <a target="_Loday"
href="aboutFunctorial.html#Milgram">counter-example</a> to a
conjecture of
J.-L. Loday. Furthermore, we get the new result that H5(M23,Z)
= Z7. 
 </li>
 <li>The Mathieu simple group M24 (of order
244823040)
has H3(M24,Z) = Z12 and H4(M24,Z)
= 0. </li>
 </ul>
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
GroupHomology(AlternatingGroup(7),10);time; 
[ 2, 3, 3, 3 ] 
1756 
 
gap> S:=Image(IsomorphismPermGroup(SL(3,3)));; 
gap> GroupHomology(S,8);time; 
[ 2, 3 ] 
6340 
 
gap> B5:=[[1,[2,3]],[2,[3,3]],[3,[4,3]],[4,[5,4]]];; 
gap> GroupHomology(["Coxeter",D],4);time; 
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ] 
56240 
 
gap> K:=MaximalSubgroups(SylowSubgroup(SL(2,Integers mod
5^3),5))[2]; 
gap> K:=Image(IsomorphismPcGroup(K)); 
gap> GroupHomology(K,3);time; 
[ 5, 5, 5, 5, 5, 5, 125 ] 
3254 
 
gap> G:=AbelianGroup([2,4,6,8,10,12]);; 
gap> GroupHomology(G,6);time; 
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 12, 12, 12 ] 
<!OCTYPE html PUBLIC"/W3C/DTD HTML 4.01 Transitional/EN"
 >
gapcontent=/tml charsetISO-8859-1"
[ ] 
9395 GroupHomologyMathieuGroup23,3)timebrjava.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
[ ] 
157961 
gap> GroupHomology(MathieuGroup(23),4);time; 
[ ] 
276853 <java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
gap> GroupHomologyMathieuGroup23)5;; 
20639802 
 <br=t: align :(0, 0 102;>a
gap> GroupHomology(MathieuGroup(24),3);time; 
[4,3 <
3205565 
 
gap&<table
[nbsp ]]br>
 </td>
 </tr>
 <
 example calculatetheintegralhomologyH<>n<sub(201>,)
style=verticalaligntop;background-:rgb255,255 255;>java.lang.StringIndexOutOfBoundsException: Range [71, 72) out of bounds for length 71
al-"GroupHomology)/
returns the mod:FreeGroup2);; x=F.1;y=F.;;br
to. The following that the Sylow 2-subgroup P of the
Mathieu simple group M24 has 6-dimensional mod 
 . (The group P has order 1024 and the computation 
 </tr>
 <tr>
 <td
background-:rgb(255,255, 204)"gap&java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
br>
[2, ,2 2, 2 ,2 2 2 2,2, 2,2 2, 2,2,2,2,,2 2,
2, 2, 
 2, 2, 2, 2, 2, on a
2, 2, 2, 2, 
&; 2, 2, 2,2 2 2,,2 2 ,2 ,2 2, 2 ,2, 2 2,2 ,
2, 2, 2, 2, 
&,,2 ,,2,2,2, ,2 ,2 , ,,2,2,2, ,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2,2, 2, 2,<>
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,becomputed in dimensions followingcommands showthat<r
 </td>
 <
 <tr>
 <td
style> java.lang.StringIndexOutOfBoundsException: Range [45, 43) out of bounds for length 70
od H>sup,Zsub2sub)
for smallish 2-groups G using the HAPprime extension package (which
uses the <>
followingcommandscomputea presentation and Poincare series for this
ring when G is the Sylow 2-subgroup of the Mathieu group M12 </represented by a Coxeter diagram on 5 vertices, has H4(B5=(Z2</ub)12</up>
The commands use the Lyndon-Hochschild-Serre spectral java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 46
Groebner bases toThisreproducesa BrowderandJP which
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; >abelian G=2/>times;<><>&;<sub6×<>8/>times;<sub10
G:=SylowSubgroup(MathieuGroup(12),</ub>times>12/ o order46080) <sub<>GZ)
 <br/ub>280+<sub4sub)12</>(<>12)3
gapg; Mod2CohomologyRingPresentation>
Graded algebra GF(2)[ x_1, x_2, x_3, x_4, x_5, has Hsub>/sub>Msub>23/sub,) H<ub3/>Msub>23</ub>,)=
[ x_2*x_3, x_1*x_3J ' <="_Loday"
x_1*x_2*x_5+x_2*x_6, 
n;x_12x_4^2*x_2^+x_6x_4^<>
 -L , we new <>5/><>23/>Zjava.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84
x_1 TheMathieu M>24/>( order
n;*5+_ ^3java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 62
&^2, 

2x_2*3x_6x_2^x_5^2+x_6x_2x_4x_2+^<
2*x_5 ] with indeterminate degrees [ 1, 1, 1,[, ,,3]>
gapgt <>
19685 
 
gap&6340>
gap&;(G;>
(1)/-3+*1^-*1<r
gap> timebr
11757 
 </td>
 </tr>
 <tr>
 <56240 
"align: ;- rgb255 ,)"The
homology ofg;K:((L(,
following commands show that 
 <ul>
 > 99- of([17)is
Hsub><sub(L>2/([7),Z) = Z4 + Z12.
(This homology was first calculated by <a
href="http://arxiv.org/abs/math/9503230">A. Adem and N. Naffah</a>). 
 </li>
 <li>the 4-dimensional integral homology of SL3(Z) is
>4</subSsub3/()Z =sub><>&;>
 <li>the 6-dimensional integral homology of the Bianchi group SL ,22 ,2 ,,,2,2 ,2 ,2,2 ,2,2 , 2 ,
with w2=-2 is H6(SL<sub2 , 2, 2,br
 
 </li>
 </ul>
 <ul>
 <li>the classical braid group B on eight strings
(represented by a linear Coxeter diagram D with seven vertices) has
5-dimensional integral homology H&bsp;2 2 ,2 ,2 ,2 2 2 ,2 ,2 ,,,2 2,
 <li>the amalgamated product G=S5*AS4
ofthe symmetric groups S5 and S4 over the
canonical subgroup A=S3 has 5-dimensional integral homology
H5(G,Z) = 2, 2 2,2, 
 </li>
 <li>the Heisenberg group H in five complex variables (a torsion
free nilpotent group of class two) has 5-dimensional integral homology H5(H,Z)
= (Z2)43+Z6+Z132.</li>
 <>the free nilpotent group N of class 2 on four generators
has 4-dimensional integral homology H4(N,Z) = (Z3)4+Z84.
(< href=.html"> </>of

for N were first calculated in a paper by L. Lambe.)</li>
 <> 3dimensional spacewith
Hermann-Mauguin symbol "P62" has 5-dimensional integral homology H5(S,Z)
=<>2<./i
 </ul>
 (The
last three examples require the "AClib", "Polycyclic" and "nq"
packages.
HAPloaded
required..) 
 </td>
 </tr <r
 <tr>
 <td
style="br>
R:(7,100)<java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
Resolution g;java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 51
No contracting[nbsp ]>
gap> Homology(gap&;((23)5;ime<>
[ 4, 12 ] 
 
 
gap>
C:=ContractibleGcomplex("SL(3,Z)"3205565br>
gap> :FreeGResolutionC,5);>
gap&tHomologyTensorWithIntegers),4; 
 
 
gap&t; C:=ContractibleGcomplex("L(2Z[sqrt(-2)]");br
gapgt; R=FreeGResolution(C,7);; 
gap> Homology(TensorWithIntegers(R),6); 
[ 2 ] 
 
 br>
gap>
D:=[ [1,[2,3]], style=verticalalign:;-color rgb255,,255 204;java.lang.StringIndexOutOfBoundsException: Range [72, 71) out of bounds for length 75
[4,[3]&; [[,],n; [6,]&bsp];br>
gap> CoxeterDiagramDisplay(D);; 
 <div style="text-align: center;"><img alt="" src="cd.gif"
tyle"idth:150px : 115;>br>
 </div>
gap> GroupHomology(D,5);time; 
[ 3 ] 
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
<>
 
 
gapgt; S5:=ymmetricGroup(5);SetName(S5,"S5"); 
gap> S4:=SymmetricGroup(4);SetName(S4,"S4"); 
gap> A:=SymmetricGroup(3); 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 , ,2,2 ,2 , 2 2,
gt :=A,-gt)>
gap> AS4 ,2 2<>
java.lang.StringIndexOutOfBoundsException: Range [37, 38) out of bounds for length 33
gap&>
 <div style="style="vertical-aligntop;background-olor: (, 255 255;"
:90px">brjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
 </div>
gap> GroupHomology(D,5);time; 
[uses the Singular system for commutative algebra). For instance, the
22004>

 
gap that computations correct.br
[ 2, 2, 2, 2, 2, 2 /tr
2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2G:SylowSubgroup(12),);>
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, x_1x_2+x_6<>
0, 0, 0, 0, 
&;0 0,0 0,0,0,,0 0, 0 0,0 0 0,0,0,0 , 0, 0,
0, 0, 0, 0, 
&bsp 0,0,0, 0 ,0,0,0 ,0 0 0, 0,,0 0,0,0 0, 0,
0, 0, 0, 0, 
&bsp;0 0, ,0,0, 0,0,0 0,0,0 ,0 , 0 , ,0 ,0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0,0,0, 0,,<r
 0 ] 
73765 
 
 
gap> F:=FreeGroup(4gap>time; 
 <r>
[, 3 0,0, 0, ,0 ,0,0, 0, 0,0, 0,0,0, , 0 0,0 0java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
0, 0,<brgap>time;<r
 0, 0, 0, 0, 0, 0, 0, 0, 0
0, 0, ,0 
&, 0, 0, 0, 0, 0, 0 ,0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 10
41967 
 
 
gap> GroupHomology(SpaceGroupBBNWZ("P62"),5);timeT was first calculated a
[ 2, 2 ] 
4336 
 </td>
 </tr>
 <tr>
 <td
style=color: rgb(255, 255, );"The
command 4SL3(Z),Z) = Z2. </li>
is a composite of several more basic HAP functions and
attempts, in a fairly crude way, to make reasonable choices for 
number of parameters in
the calculation of group homologyrthe java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 56
almost
certainly be better off using the more basic functions directly and
making the
choices yourself! Similar comments apply to functions for cohomology
(ring) calculations. 
 
The subsequent pages of this manual explain the basic HAP functions. The
intending reader should be aware that many S5/> sub<> the
java.lang.StringIndexOutOfBoundsException: Range [28, 29) out of bounds for length 2
the fullpotential of HAPandnbsp consequently may take
many minutes (and in one or two cases product can be represented as a graph of groups.) 
 </java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255 li>the free nilpotent group N of class 2 on four generators
p S the HAPcryst extension(hich
uses the CrystGAPpackage thePolymake geometry
system) can be used to for werefirst ina paperbyL. Lambe)/>
java.lang.StringIndexOutOfBoundsException: Range [35, 5) out of bounds for length 68
For instance, the following commands compute a fundamental cell for the
-dimensionalspacegroup with
Hermann-Mauguin symbol "P62" and exhibit the 1-skeleton of this cell. 
 </td>
 </ </>
 tr
 <td examples require "Clib" "olycyclic" and"qjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
style="vertical-align: top; background-color: rgb(255, 255, 204);".)<r>
fd=FundamentalDomainStandardSpaceGroup,315,("P62");<>
gap&t;&bsp;Polymake(fd"VISUAL_GRAPH";<>
 
 div style=textalign center"<img
style="width: 300px; height: 300px;" alt="" src="Fundom.png"> 
 </div>
 </td>
 </tr>
 <tr>
 td
style=" 
end this java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 7
calculations such as: 
 uljava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
 <li>The rank of [2]br
of the Mathieu group M11 gap> C:=ContractibleGcomplex("SL(2,(-2)]));;<r>
coefficient of xk in the Poincare java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 37
for all kless than 15 (ThisPoincare series for the ring H*(M11,Z2)
was first calculated in [P.Webb, "A java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 7
style="ont-style:italic;">Comm. Math. Helv. 62 (1987)
135-167]. ) </li>
 <li>The mod 2 cohomology4,5,3]&; 5[,],java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 57
for the dihedral group of order 64 is generateddiv styletalign center"<img alt="" src="cd.gif"
degree 1 and one element of <div>>
likely)
some generators of degree greater than 30. 
 </li>
 <li>The Lie algebra M3(Z)g; S5=SymmetricGroup5)SetName) 
matrices has 5-> A:=SymmetricGroup(A," 
 <li>The suspension X=SK(G,gap& :=GroupHomomorphismByFunction(,5,-g;) 
pi<sub&; GraphOfGroupsDisplay() 
 liThedoublesuspension=(,1 ofan - Lane
space for the group G=GL(4,3) of 4×4 matrices over the field of
three elements (of order 24261120) has fourth homotopy gapg;GroupHomologyD5;ime<>
= Z2 . 
 </li>
 <li>The free nilpotent Lie algebra A ofg;GroupHomologyHeisenbergPcpGroup))t; 
generators, over,2, ,2 2 2 2 ,2 2,,2 ,2 22,,2,
the ring of integers Z, has 3-dimensional Leibniz homology HL3</subnbsp;; 2,2,2, 2, 2, 2, 2 2,2 2, 2, ,2 2,2 2,2 2 2, 2, 6,0 0java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
+ (Z6)16 +Z176 .</li>
 <li>Thegroup presentation = <,y,z,bc, |axy
b=yz, c=zx, ax=ya, by=zb, cz=xc > is aspherical.</li>
 <li>The 3-dimensional module M over the field F of two
 elements, arising from the canonical left action of the group G=Syl2(GL3(2))
of 3t;3matrices(order8),hasa 6-imensional Ext
module Ext5FG(M,F)=F6.</li>
 <li>The 3-dimensional integral homology of the
homotopy 2-type X represented by the automorphism crossed module D16
--> Aut(D16) is H3(X,Z)=Z2+Z2+Z4.
 
 </ul>
The following commands yield these seven calculations. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 9
PoincareSeriesPrimePart(MathieuGroup(11),2,14); br
(x^4-x^3+x^2-x+1)/(x^6-x^5+x^4-2*x^3+x^2-x+1) 
 
gap> H:=ModPCohomologyGenerators(DihedralGroup(64),30);; 
gap> List(H[1], H[2]); 
[ 0, 1, 1, 2 ] 
 
gap>
A:=MatLieAlgebra(Integers,3);; 
gap> LieAlgebraHomology(A,5); 
java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 33
 
gap> F:=FreeGroup(4);;G:=NilpotentQuotient(F,2);; 
gap> ThirdHomotopyGroupOfSuspensionB(G); 
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 
 0, 0, 0, 0, 0 ] 
 
gap&t G:Image(IsomorphismPermGroup(GL43); 
gap> NonabelianSymmetricKernel_alt(G); 
[ [ ], [ 2 ] ] 
 
gap> F:=FreeGroup(4);;G:=NilpotentQuotient(F,2);; 
gt :=LowerCentralSeriesLieAlgebra(G; 
gap> LeibnizAlgebraHomology(L,3); 
[ 2, 2, 2, 2, 2, 2, 2, 2, 6, 6, 6, 6,
6, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 6
0, 0T subsequentpagesof thismanualexplainthe basic functions <span<>The
&bsp , 0 ,0, 0 0,0 0,,0 ,0, 0 , 0 0 0,0,,0 0java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, minutesand cases run</>/java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, style"-java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 71
0, 0,0,0,>
&, 0,0 0,,0,0,0,0 0 0,0,0, 0, 0, 0, 0, 0, 0, 0,
0,0,0,0,<r
 0, 0, 0, 0, 0, 0, 0, 0, 0<>
 ,0<>
 
g;
java.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 69
gap> rels:=[a^-1*x*y, b^-1*y*z, c^-1*z*x, a*x*(y*a)^-1,
b*y*(z*b)^-1, c*z*(x*c)^-1];; 
gap> IsAspherical(F,rels);; 
Presentation is aspherical. 
 
gap> M:=GModuleByMats(GeneratorsOfGroup(SylowSubgroup(GL(3,2)<
gap> R:=ResolutionFpGModule(DesuspensionMtxModule(M),5);; 
gap> Cohomology(HomToIntegersModP(R,2),4); 
6 

gap& theMathieu sub order7920 is to the
C:AutomorphismGroupAsCatOneGroup(DihedralGroup(32)); 
gapfor all k than15.( Poincare for ringHsup*<sub,sub2<subjava.lang.StringIndexOutOfBoundsException: Index 102 out of bounds for length 102
gap>
K:=ChainComplexOfSimplicialGroup(N);; 
gap135-167] /li>
[ 2 ,4 ]<>
 </td>
 <>
 <tr>
 <td style=verticalalign top"java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
 <table style="width: 100%; text-align: left;" border="0"
cellpadding="" cellspacing""
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