The main purpose of this page is to provide usable generators and relations for all groups of order 32. The main reason that I created this was that
it suited my needs and it was hard to find a page with generators and relations the way I want them. Too many times, I got frustrated asking GAP for
the derived subgroup of a particular group and getting back the subgroup generated by "f4" and "f5", especially when the group has a very easy
2-generator presentation. The format is as follows.
The GAP number of the group is listed as well as the output of "StructureDescription" in GAP.
In a few cases, I have added a secondary description of the group. Since I had to ask someone about what the notation meant, I'm guessing other
people wonder the same thing. A group G is an extension of A by B if A is a normal subgroup of G and G/A is isomorphic to B. This gives a natural
exact sequence 1 --> A --> G --> B --> 1. This extension is said to be split if you can find a map from B to G that is consistent with the maps in
the exact sequence. This says that G actually has a subgroup isomorphic to B, so G is a semi-direct product of A and B. This is written as A:B. If
the extension is not split, we write A.B
A free group is constructed. The generators are listed alphabetically, starting from "a". Direct factors always come last as do secondary factors in extensions, both split and non-split. I typically tried
to present things in power-commutator format. I am not using the minimal number of generators since it is frankly less confusing, at least to me, if
the relations are minimal and don't contain words of long length.
To use these, copy and paste everything from the definition of the free group to the definition of g into GAP.
Here they are:
GAP group 1
Structure Description: Z_32
f:=FreeGroup("a");
a:=f.1;
g:=f/[a^32];
GAP group 2
Structure Description: (Z4 x Z2): Z4
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^4,b^2,c^4,Comm(a,b),a^c*a^(-1)*b,Comm(b,c)];
GAP group 3
Structure Description Z8 x Z4
f:=FreeGroup("a","b");
a:=f.1;
b:=f.2;
g:=f/[a^8,b^4,Comm(a,b)];
GAP group 4
Structure Description Z8 : Z4
f:=FreeGroup("a","b");
a:=f.1;
b:=f.2;
g:=f/[a^8,b^4,Comm(a,b)*a^4];
GAP group 5
Structure Description (Z8 x Z2) : Z2
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^8,b^2,c^2,Comm(a,b),a^c*a^(-1)*b,Comm(b,c)];
GAP group 6
Structure Description ((Z4 x Z2) : Z2) : Z2
f:=FreeGroup("a","b","c","d");
a:=f.1;
b:=f.2;
c:=f.3;
d:=f.4;
g:=[a^4,b^2,c^2,d^2,Comm(a,b),a^c*a^(-1)*b,Comm(b,c),a^d*a^(-1)*c,Comm(b,d),Comm(c,d)];
GAP group 7
Structure Description (Z8 : Z2) : Z2
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^8,b^2,c^2,a^b*a^3,a^c*a^(-1)*b,Comm(b,c)];
GAP group 8
Structure Description Z2 . ((Z4 x Z2) : Z2) = (Z2 x Z2) . (Z4 x Z2)
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^8,b^2,c^2*a^4,a^b*a^3,a^c*b*a^(-1),Comm(b,c)];
GAP group 9
Structure Description (Z8 x Z2) : Z2
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^8,b^2,c^2,Comm(a,b),a^c*a*b,Comm(b,c)];
GAP group 10
Structure Description Q8 : Z4
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^4,b^4,c^4,a^2*b^2,a^b*a,a^c*b^(-1),b^c*a^(-1)];
GAP group 11
Structure Description (Z4 x Z4) : Z2
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^4,b^4,c^2,Comm(a,b),a^c*b^(-1),b^c*a^(-1)];
GAP group 12
Structure Description Z4 : Z8
f:=FreeGroup("a","b");
a:=f.1;
b:=f.2;
g:=f/[a^4,b^8,Comm(a,b)*a^2];
GAP group 13
Structure Description Z8 : Z4
f:=FreeGroup("a","b");
a:=f.1;
b:=f.2;
g:=f/[a^8,b^4,Comm(a,b)*a^6];
GAP group 14
Structure Description Z8 : Z4
f:=FreeGroup("a","b");
a:=f.1;
b:=f.2;
g:=f/[a^8,b^4,Comm(a,b)*a^2];
GAP group 15
Structure Description Z4 . D8 = Z4 . (Z4 x Z2)
f:=FreeGroup("a","b");
a:=f.1;
b:=f.2;
g:=f/[a^8,b^8,a^b*a,a^4*b^4];
GAP group 16
Structure Description Z16 x Z2
f:=FreeGroup("a","b");
a:=f.1;
b:=f.2;
g:=f/[a^16,b^2,Comm(a,b)];
GAP group 17
Structure Description Z16 : Z2
f:=FreeGroup("a","b");
a:=f.1;
b:=f.2;
g:=f/[a^16,b^2,Comm(a,b)*a^8];
GAP group 18
Structure Description D32
f:=FreeGroup("a","b");
a:=f.1;
b:=f.2;
g:=f/[a^16,b^2,Comm(a,b)*a^2];
GAP group 19
Structure Description QD32
f:=FreeGroup("a","b");
a:=f.1;
b:=f.2;
g:=f/[a^16,b^2,a^b*a^9];
GAP group 20
Structure Description Q32
f:=FreeGroup("a","b");
a:=f.1;
b:=f.2;
g:=f/[a^16,b^4,a^b*a,a^8*b^2];
GAP group 21
Structure Description Z4 x Z4 x Z2
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^4,b^4,c^2,Comm(a,b),Comm(a,c),Comm(b,c)];
GAP group 22
Structure Description ((Z4 x Z2) : Z2) x Z2
f:=FreeGroup("a","b","c","d");
a:=f.1;
b:=f.2;
c:=f.3;
d:=f.4;
g:=[a^4,b^2,c^2,d^2,Comm(a,b),a^c*a^(-1)*b,Comm(b,c),Comm(a,d),Comm(b,d),Comm(c,d)];
GAP group 23
Structure Description (Z4 : Z4) x Z2
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^4,b^4,c^2,a^b*a,Comm(a,c),Comm(b,c)];
GAP group 24
Structure Description (Z4 x Z4) : Z2
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^4,b^4,c^2,Comm(a,b),a^c*a^(-1)*b^2,Comm(b,c)];
GAP group 25
Structure Description D8 x Z4
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^4,b^2,c^4,Comm(a,c),Comm(b,c),a^b*a];
GAP group 26
Structure Description Q8 x Z4
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^4,b^4,c^4,Comm(a,c),Comm(b,c),a^b*a,a^2*b^2];
GAP group 27
Structure Description (Z2 x Z2 x Z2 x Z2) : Z2
f:=FreeGroup("a","b","c","d","e");
a:=f.1;
b:=f.2;
c:=f.3;
d:=f.4;
e:=f.5;
g:=f/[a^2,b^2,c^2,d^2,e^2,Comm(a,b),Comm(a,c),Comm(a,d),a^e*a*b,Comm(b,c),Comm(b,d),Comm(b,e),Comm(c,d),c^e*c*d,Comm(d,e)];
GAP group 28
Structure Description (Z4 x Z2 x Z2) : Z2
f:=FreeGroup("a","b","c","d");
a:=f.1;
b:=f.2;
c:=f.3;
d:=f.4;
g:=f/[a^4,b^2,c^2,d^2,Comm(a,b),Comm(a,c),Comm(b,c),a^d*a,b^d*b*c,Comm(c,d)];
GAP group 29
Structure Description (Q8 x Z2) : Z2
f:=FreeGroup("a","b","c","d");
a:=f.1;
b:=f.2;
c:=f.3;
d:=f.4;
g:=f/[a^4,b^4,a^2*b^2,a^b*a,c^2,Comm(a,c),Comm(b,c),d^2,Comm(a,d)*c,Comm(b,d),Comm(c,d)];
GAP group 30
Structure Description (Z4 x Z2 x Z2) : Z2
f:=FreeGroup("a","b","c","d");
a:=f.1;
b:=f.2;
c:=f.3;
d:=f.4;
g:=f/[a^4,b^2,c^2,d^2,Comm(a,b),Comm(a,c),Comm(b,c),a^d*b*a^(-1),Comm(b,d),Comm(c,d)*a^2];
GAP group 31
Structure Description (Z4 x Z4) : Z2
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^4,b^4,c^2,Comm(a,b),a^c*a*b^2,b^c*b];
GAP group 32
Structure Description (Z2 x Z2) . (Z2 x Z2 x Z2)
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^4,b^4,a^b*a,c^4,a^2*c^2,Comm(a,c)*b^2,Comm(b,c)];
GAP group 33
Structure Description (Z4 x Z4) : Z2
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^4,b^4,c^2,Comm(a,b),a^c*a^(-1)*b^2,b^c*b*a^2];
GAP group 34
Structure Description (Z4 x Z4) : Z2
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^4,b^4,c^2,Comm(a,b),a^c*a,b^c*b];
GAP group 35
Structure Description Z4 : Q8
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^4,b^4,c^4,b^2*c^2,b^c*b,Comm(a,b)*a^2,Comm(a,c)];
GAP group 36
Structure Description Z8 x Z2 x Z2
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^8,b^2,c^2,Comm(a,b),Comm(a,c),Comm(b,c)];
GAP group 37
Structure Description (Z8 : Z2) x Z2
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^8,b^2,c^2,a^b*a^3,Comm(a,c),Comm(b,c)];
GAP group 38
Structure Description (Z8 x Z2) : Z2
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^8,b^2,c^2,Comm(a,b),Comm(a,c),b^c*b*a^4];
GAP group 39
Structure Description D16 x Z2
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^8,b^2,c^2,Comm(a,c),Comm(b,c),a^b*a];
GAP group 40
Structure Description QD16 x Z2
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^8,b^2,c^2,Comm(a,c),Comm(b,c),a^b*a^5];
GAP group 41
Structure Description Q16 x Z2
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^8,b^4,c^2,Comm(a,c),Comm(b,c),a^b*a,a^4*b^2];
GAP group 42
Structure Description (Z8 x Z2) : Z2
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^8,b^2,c^2,Comm(a,b),a^c*a,b^c*b*a^4];
GAP group 43
Structure Description (D8 x Z2) : Z2
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^8,b^2,c^2,a^b*a^3,a^c*a^5,Comm(b,c)];
GAP group 44
Structure Description (Z2 x Q8) : Z2
f:=FreeGroup("a","b","c");
a:=f.1;
b:=f.2;
c:=f.3;
g:=f/[a^8,b^2,c^2,a^b*a^3,a^c*a^5,b^c*b*a^4];
GAP group 45
Structure Description Z4 x Z2 x Z2 x Z2
f:=FreeGroup("a","b","c","d");
a:=f.1;
b:=f.2;
c:=f.3;
d:=f.4;
g:=f/[a^4,b^2,c^2,d^2,Comm(a,b),Comm(a,c),Comm(a,d),Comm(b,c),Comm(b,d),Comm(c,d)];
GAP group 46
Structure Description D8 x Z2 x Z2
f:=FreeGroup("a","b","c","d");
a:=f.1;
b:=f.2;
c:=f.3;
d:=f.4;
g:=f/[a^4,b^2,c^2,d^2,a^b*a,Comm(a,c),Comm(a,d),Comm(b,c),Comm(b,d),Comm(c,d)];
GAP group 47
Structure Description Q8 x Z2 x Z2
f:=FreeGroup("a","b","c","d");
a:=f.1;
b:=f.2;
c:=f.3;
d:=f.4;
g:=f/[a^4,b^4,c^2,d^2,a^b*a,a^2*b^2,Comm(a,c),Comm(a,d),Comm(b,c),Comm(b,d),Comm(c,d)];
GAP group 48
Structure Description ((Z4 x Z2) : Z2) x Z2
f:=FreeGroup("a","b","c","d");
a:=f.1;
b:=f.2;
c:=f.3;
d:=f.4;
g:=f/[a^4,b^2,c^2,d^2,Comm(a,b),Comm(a,d),Comm(b,d),a^c*a,Comm(b,c)*a^2,Comm(c,d)];
GAP group 49
Structure Description (D8 x Z2) : Z2 or a central product of D8 with D8
f:=FreeGroup("a","b","c","d");
a:=f.1;
b:=f.2;
c:=f.3;
d:=f.4;
g:=f/[a^4,b^2,c^4,d^2,a^b*a,Comm(a,c),Comm(a,d),Comm(b,c),Comm(b,d),c^d*c,a^2*c^2];
GAP group 50
Structure Description (Q8 x Z2) : Z2 or a central product of D8 with Q8
f:=FreeGroup("a","b","c","d");
a:=f.1;
b:=f.2;
c:=f.3;
d:=f.4;
g:=f/[a^4,b^2,c^4,d^4,a^b*a,Comm(a,c),Comm(a,d),Comm(b,c),Comm(b,d),c^d*c,c^2*d^2,a^2*c^2];
GAP group 51
Structure Description Z2 x Z2 x Z2 x Z2 x Z2
f:=FreeGroup("a","b","c","d","e");
a:=f.1;
b:=f.2;
c:=f.3;
d:=f.4;
e:=f.5;
g:=f/[a^2,b^2,c^2,d^2,e^2,Comm(a,b),Comm(a,c),Comm(a,d),Comm(a,e),Comm(b,c),Comm(b,d),Comm(b,e),Comm(c,d),Comm(c,e),Comm(d,e)];