Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4×C8 · Schur multiplier (H2) = C2×C3×C4 |
| direct product | C32×#1281 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | C22⋊C4 |
| commutator | C22 | C32×C4×C8 |
| Frattini, cosocle | C2×C42 | C22×C32 |
| socle | C22×C32 | #95 |
| prime radical | C23×C32 | C42 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ C2×C32×C42 ◁ G | |
456 conjugacy classes of subgroups; only the 264 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 264 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) = 576 |
A minimal faithful permutation representation — 2 generators on 30 points:
(1,2,3)(7,8,9,11,10,12,13,14)(15,16)(17,23,24,20)(18,22)(19,21)(25,29,30,27)(26,28);(4,5,6)(15,17,18,25,19,24,26,30)(16,20,22,27,21,23,28,29)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2341);