Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×S4×GO+(4,𝔽2) |
| Fitting, socle | C23×C32 ⋊ D3×D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | S4×GO+(4,𝔽2) |
| commutator, cosocle | A4×C3⋊D3 | C24 |
| Frattini | C1 | G |
| Fitting, socle | C23×C32 | D3×D4 |
| hypocenter | C32×A4 | C22×D4 |
| LCS | G ▷ A4×C3⋊D3 ▷ C32×A4 | |
| UCS | C1 ◁ C2 | |
4120 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 4120 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 4 generators on 12 points:
(3,6,5)(7,10,11,8)(9,12);(10,12);(1,2);(4,6)90 conjugacy classes — view the full character table.
| maximal subgroup of | C2×S4×AΓL(1,𝔽9) |
| maximal normal subgroup of | C2×S4×AΓL(1,𝔽9) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,6,5)(7,10,11,8)(9,12), (10,12), (1,2), (4,6));