Finite Group DatabasebetaFinite Group Databasebeta| order | 2016 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×A4×S4 |
| Fitting, socle | C24×C7 ⋊ C3×D3 |
| derived | C22×A4 ⋊ C2×C3×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | A4×S4 |
| commutator, cosocle, hypocenter | C22×A4 | C2×C3×C7 |
| Frattini | C1 | G |
| Fitting, socle | C24×C7 | C3×D3 |
| LCS | G ▷ C22×A4 | |
| UCS | C1 ◁ C7 | |
172 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 15 points:
(2,4,3)(5,8,6,7)(9,10,11,12,13,14,15);(1,4)(2,3)(6,7,8)140 conjugacy classes — view the full character table.
| maximal subgroup of | A4×ΓL(1,𝔽8)×S4, C7×S42, A4×D7×S4, … |
| maximal normal subgroup of | A4×ΓL(1,𝔽8)×S4, C7×S42, A4×D7×S4, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,4,3)(5,8,6,7)(9,10,11,12,13,14,15), (1,4)(2,3)(6,7,8));