Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C2×C4 |
| direct product | C9×#1921 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C2×D4 |
| commutator | C2×C4 | C22×C4×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle, prime radical | C23×C3 | C2×C3×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ C2×C42×C9 ◁ G | |
345 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5,6,7,8,9)(10,11,13,12)(14,15)(16,19)(17,18)(20,21)(22,23,24,26)(25,27,29,28);(10,12,13,11)(15,18)(19,21)(23,26)(27,28);(14,16,17,20)(15,19,18,21)(23,27)(25,29)(26,28)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 653);