Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C9 · Schur multiplier (H2) = C24 |
| direct product | C9×#2382 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C24 |
| commutator | C22 | C23×C4×C9 |
| Frattini, socle, prime radical, cosocle | C23×C3 | C24×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ G | |
612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5,6,7,8,9)(10,11,12,13)(14,15)(16,20)(17,18)(19,21)(23,24)(27,28);(15,18)(16,21)(22,23,26,24)(25,28,29,27);(14,15)(16,19)(17,18)(20,21)(22,24,26,23)(25,28,29,27);(14,16,17,21)(15,19,18,20)(22,25,26,29)(23,27,24,28)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1114);