Finite Group DatabasebetaFinite Group Databasebeta| order | 8640 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C2×C3 |
| direct product | SL(2,𝔽3)×A6 |
| Fitting | Q8 ⋊ C3×A6 |
| derived | Q8×A6 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4×A6 |
| commutator, hypocenter | Q8×A6 | C3 |
| Fitting, cosocle | Q8 | C3×A6 |
| socle | C2×A6 | A4 |
| LCS | G ▷ Q8×A6 | |
| UCS | C1 ◁ C2 | |
236 conjugacy classes of subgroups; only the 8 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 8 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,6,8)(2,4,7)(9,10,11,14,12);(1,4,2,6)(3,7,5,8)(9,11)(10,14,13,12)49 conjugacy classes — view the full character table.
| maximal subgroup of | SL(2,𝔽5)×A6 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,6,8)(2,4,7)(9,10,11,14,12), (1,4,2,6)(3,7,5,8)(9,11)(10,14,13,12));