Conjugacy classes and centralizers of a SmallGroup

Question

• What is the complete lists of conjugacy classes and centralizers of SmallGroup(64,138)?

Would someone be willing to provide the complete lists of conjugacy classes and centralizers of SmallGroup(64,138)?

I had heard the GAP program helps, but I tried for a while and have troubles to install it in my computer. Thank you. :o)

Answer 1

This is really straightforward calculation in GAP:

gap> G:=SmallGroup(64,138);
<pc group of size 64 with 6 generators>
gap> StructureDescription(G);
"(((C4 x C2) : C2) : C2) : C2"
gap> cc:=ConjugacyClasses(G);
[ <identity> of ...^G, f1^G, f2^G, f3^G, f4^G, f5^G, f6^G, f1*f2^G, f1*f3^G, 
  f1*f6^G, f2*f3^G, f2*f5^G, f3*f4^G, f4*f5^G, f1*f2*f3^G, f2*f3*f4^G ]
gap> List(cc,Centralizer);
[ (((C4 x C2) : C2) : C2) : C2, Group([ f1, f4, f5, f6 ]), Group([ f2, f3, f4, f6     ]),
  Group([ f2, f3, f5, f6 ]), Group([ f1, f2, f4, f5, f6 ]), 
  Group([ f1, f3, f4, f5, f6 ]), Group([ f1, f2, f3, f4, f5, f6 ]), 
  Group([ f1*f2, f4, f6 ]), Group([ f1*f3, f5, f6 ]), Group([ f1, f4, f5, f6 ]), 
  Group([ f2, f3, f4*f5, f6 ]), Group([ f2*f5, f3, f4, f6 ]), 
  Group([ f2, f3*f4, f5, f6 ]), Group([ f1, f2*f3, f4, f5, f6 ]), 
  Group([ f1*f2*f3, f4*f5, f6 ]), Group([ f2, f3*f5, f4*f5, f6 ]) ]

but perhaps its output in this form makes little sense outside GAP, since you have to know what are the generators and relations. The group is given by the polycyclic presentation, and its defining relations may be seen here:

gap> H:=Image(IsomorphismFpGroup(G));
<fp group of size 64 on the generators [ F1, F2, F3, F4, F5, F6 ]>
gap> RelatorsOfFpGroup(H);
[ F1^2, F2^-1*F1^-1*F2*F1*F4^-1, F3^-1*F1^-1*F3*F1*F5^-1, F4^-1*F1^-1*F4*F1, 
  F5^-1*F1^-1*F5*F1, F6^-1*F1^-1*F6*F1, F2^2, F3^-1*F2^-1*F3*F2, F4^-1*F2^-1*F4*F2, 
  F5^-1*F2^-1*F5*F2*F6^-1, F6^-1*F2^-1*F6*F2, F3^2, F4^-1*F3^-1*F4*F3*F6^-1, 
  F5^-1*F3^-1*F5*F3, F6^-1*F3^-1*F6*F3, F4^2, F5^-1*F4^-1*F5*F4, F6^-1*F4^-1*F6*F4, 
  F5^2, F6^-1*F5^-1*F6*F5, F6^2 ]

Hope this helps. If you have a technical question about GAP installation, it would be off-topic here. For such kind of advice, please contact GAP Support and tell which operating system do you use and what is the problem.