Finite Group DatabasebetaFinite Group Databasebeta| order | 12996 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C3×C19 |
| unambiguous semidirect product | C19⋊(C19⋊#157) |
| Fitting, socle, derived, Hall | C32×C192 ⋊ C4 |
| Sylow | C192 ⋊ #157 |
| Sylow | C32 ⋊ C19⋊Dic19 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C32×C192 | C4 |
| prime radical, cosocle | C3⋊D3×C192 | C2 |
| LCS | G ▷ C32×C192 | |
| UCS | C1 | |
220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 2166 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2)(3,5,6,4)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44);(1,3,6)(2,5,4)(26,27,29,31,33,35,37,39,41,43,44,42,40,38,36,34,32,30,28);(7,8,10,12,14,16,18,20,22,24,25,23,21,19,17,15,13,11,9)Polycyclic presentation:
2^2,3^2,19^2|a^2=b,c^a=cd,d^a=cd^2,e^a=e^18,f^a=f^18,c^b=c^2,d^b=d^21086 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12996, 288);