Finite Group DatabasebetaFinite Group Databasebeta| order | 2160 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| direct product | D32×A5 |
| Fitting | C32 ⋊ C22×A5 |
| socle, derived | C32×A5 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, hypocenter | C32×A5 | C22 |
| Fitting, cosocle | C32 | C22×A5 |
| LCS | G ▷ C32×A5 | |
| UCS | C1 | |
340 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 340 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:
(2,3)(4,6,5)(7,8)(9,11);(1,2)(4,6)(8,10,9)45 conjugacy classes — view the full character table.
| maximal subgroup of | D3×A52, 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((2,3)(4,6,5)(7,8)(9,11), (1,2)(4,6)(8,10,9));