Finite Group DatabasebetaFinite Group Databasebeta| order | 2160 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×D3×S5 |
| Fitting | C32 ⋊ C2×S5 |
| socle | C32×A5 ⋊ C22 |
| derived | C3×A5 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D3×S5 |
| commutator, cosocle, hypocenter | C3×A5 | C22×C3 |
| Frattini | C1 | G |
| Fitting | C32 | C2×S5 |
| socle | C32×A5 | C22 |
| LCS | G ▷ C3×A5 | |
| UCS | C1 ◁ C3 | |
281 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 281 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 30 |
A minimal faithful permutation representation — 2 generators on 11 points:
(1,2)(4,5,6)(7,9,8);(1,3,2)(7,9)(8,10,11)63 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#72219548, D32×S5 |
| maximal normal subgroup of | D32×S5 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(4,5,6)(7,9,8), (1,3,2)(7,9)(8,10,11));