Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C2×C3×C4 |
| direct product | C32×#1556 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C32×C4 |
| Frattini, cosocle | C42 | C23×C32 |
| prime radical | C23×C32 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C2×C32×C42 ◁ G | |
684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2,3)(7,8,10,12,11,13,9,14)(15,16,18,22,19,21,27,29)(17,23,24,30,25,20,26,28);(4,5,6)(8,12)(9,10)(13,14)(15,17,19,25)(16,20,21,23)(18,26,27,24)(22,30,29,28);(8,13)(12,14)234 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2616);