Finite Group DatabasebetaFinite Group Databasebeta| order | 4320 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Fitting | C22×C3 ⋊ ΓL(2,𝔽4) |
| socle | C22×C3×A5 ⋊ D3 |
| derived | C3×A4×A5 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×A4×A5 | C2 |
| Fitting | C22×C3 | ΓL(2,𝔽4) |
| socle | C22×C3×A5 | D3 |
| LCS | G ▷ C3×A4×A5 | |
| UCS | C1 | |
451 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 451 conjugacy classes of subgroups — too many to draw. Only its 17 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) = 60 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2,3);(4,7,5)(8,10,12);(1,2)(6,7)(9,11,12,10)39 conjugacy classes — view the full character table.
| maximal subgroup of | #72247654, D3×A4⋊S5 |
| maximal normal subgroup of | D3×A4⋊S5 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3), (4,7,5)(8,10,12), (1,2)(6,7)(9,11,12,10));