Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#10596770⋊C2 |
| Fitting, socle | C24×C3 ⋊ GO+(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #10596770⋊C2 |
| commutator, cosocle | A4⋊S4 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C24×C3 | GO+(4,𝔽2) |
| hypocenter | A42 | C3×D4 |
| LCS | G ▷ A4⋊S4 ▷ A42 | |
| UCS | C1 ◁ C3 | |
476 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 476 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) = 24 |
A minimal faithful permutation representation — 2 generators on 11 points:
(1,4,3)(6,8)(9,10,11);(1,2)(3,6)(4,5)(7,8)60 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#10596770⋊C2, C2×C3×#10596770⋊C2 |
| maximal normal subgroup of | D3×#10596770⋊C2, C2×C3×#10596770⋊C2 |
| maximal quotient of | C2×C3×#10596770⋊C2 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,3)(6,8)(9,10,11), (1,2)(3,6)(4,5)(7,8));