Finite Group DatabasebetaFinite Group Databasebeta| order | 12096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| direct product | S4×A4⋊AGL(1,𝔽7) |
| Fitting, socle | C24×C7 ⋊ C3×D32 |
| derived | C7×A42 ⋊ C22×C3 |
| Sylow | C7 ⋊ C3×S42 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C7×A42 | C22×C3 |
| Fitting, socle | C24×C7 | C3×D32 |
| LCS | G ▷ C7×A42 | |
| UCS | C1 | |
1202 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 46 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:
(2,6,5,3,4,7)(8,11,9,10)(12,14,15,13);(1,2,5,6,4,7,3)(9,10,11)(14,15)95 conjugacy classes — view the full character table.
| maximal subgroup of | S42×AGL(1,𝔽7) |
| maximal normal subgroup of | S42×AGL(1,𝔽7) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,6,5,3,4,7)(8,11,9,10)(12,14,15,13), (1,2,5,6,4,7,3)(9,10,11)(14,15));