Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×#2099 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C32 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23×C32 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ C2×C32×C42 ◁ G | |
642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2,3)(7,8,11,13)(9,14,10,12)(15,16)(17,19)(18,20)(21,22);(4,5,6)(7,9,11,10)(8,12,13,14)(15,17,18,21)(16,19,20,22)(23,24)(25,27)(26,29)(28,30);(7,10,11,9)(8,12,13,14)(16,20)(19,22)(24,27)(25,30)(28,29)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3159);