Finite Group DatabasebetaFinite Group Databasebeta| order | 2160 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| direct product | D3×ΓL(2,𝔽4) |
| Fitting | C32 ⋊ C2×S5 |
| socle, derived | C32×A5 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, cosocle, hypocenter | C32×A5 | C22 |
| Fitting | C32 | C2×S5 |
| LCS | G ▷ C32×A5 | |
| UCS | C1 | |
303 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 303 conjugacy classes of subgroups — too many to draw. Only its 16 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:
(1,3)(5,6)(8,11);(1,2,3)(4,5)(7,11,8,9,10)36 conjugacy classes — view the full character table.
| maximal subgroup of | D3×AΓL(2,𝔽4), #72247654, D3×A4⋊S5, … |
| maximal normal subgroup of | D32×S5 |
| maximal quotient of | D3×AΓL(2,𝔽4), D3×A4⋊S5 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,3)(5,6)(8,11), (1,2,3)(4,5)(7,11,8,9,10));