Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C8 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×#1272 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C32 | C22⋊C4 |
| commutator | C2×C4 | C2×C32×C8 |
| Frattini, cosocle | C2×C42 | C22×C32 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ C2×C32×C42 ◁ G | |
498 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 498 conjugacy classes of subgroups — too many to draw. Only its 204 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 2 generators on 30 points:
(1,2,3)(7,8)(9,13,14,11)(10,12)(16,19,20,21)(23,24,25,27,26,28,29,30);(4,5,6)(7,9)(8,11)(10,14)(12,13)(15,16,18,20)(17,21,22,19)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2332);