Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C22×C3×C4 |
| direct product | C32×#1846 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C32 | C2×D4 |
| commutator | C2×C4 | C22×C32×C4 |
| Frattini, cosocle | C22×C4 | C23×C32 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ C2×C32×C42 ◁ G | |
744 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)(8,10)(11,12,13,14)(15,16)(17,23)(18,26)(19,21)(20,24)(22,27)(25,29)(28,30);(4,5,6)(15,17,18,24,19,25,27,30)(16,20,22,23,21,28,26,29);(7,8,9,10)(16,21)(20,28)(22,26)(23,29)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2906);