Finite Group DatabasebetaFinite Group Databasebeta| order | 12996 |
| automorphism | ? |
| homology | Abelianization (H1) = C9 · Schur multiplier (H2) = C2×C19 |
| unambiguous semidirect product | C22⋊#15692430 |
| derived, Hall | C22×C192 ⋊ C9 |
| Sylow | C192 ⋊ C22⋊C9 |
| Sylow | C22 ⋊ #15692430 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | C22⋊#11762246 |
| commutator, hypocenter | C22×C192 | C9 |
| Fitting, socle, prime radical, cosocle | C22×C3×C192 | C3 |
| LCS | G ▷ C22×C192 | |
| UCS | C1 ◁ C3 | |
92 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 342 |
A minimal faithful permutation representation — 2 generators on 51 points:
(2,3,4)(5,6,8,7,9,11,10,12,13)(15,16,18)(17,20,21)(19,23,22)(24,28,26)(25,29,27)(30,32,31)(34,35,37)(36,39,43)(38,41,40)(42,46,47)(44,48,45)(49,51,50);(1,4)(2,3)(14,15,17,21,26,31,28,16,19,24,29,18,22,27,20,25,30,32,23)(33,34,36,39,44,49,48,37,40,45,46,35,38,42,43,47,50,51,41)Polycyclic presentation:
3,2^2,19^2,3|a^3=f,b^a=c,c^a=bc,d^a=d^7,e^a=e^111452 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12996, 97);