Finite Group DatabasebetaFinite Group Databasebeta| order | 4032 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C4×S4×AGL(1,𝔽7) |
| Fitting | C22×C4×C7 ⋊ C2×C3×D3 |
| derived | C7×A4 ⋊ C22×C3×C4 |
| Sylow | C7 ⋊ C2×C3×C4×S4 |
964 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 964 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 84 |
A minimal faithful permutation representation — 3 generators on 15 points:
(5,8,7)(10,13,14,11,12,15);(1,2,3,4)(9,10,12,15,14,13,11);(6,8)140 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×S4×AGL(1,𝔽7), D4×S4×AGL(1,𝔽7) |
| maximal normal subgroup of | D4×S4×AGL(1,𝔽7) |
| maximal quotient of | AGL(1,𝔽5)×S4×AGL(1,𝔽7) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,8,7)(10,13,14,11,12,15), (1,2,3,4)(9,10,12,15,14,13,11), (6,8));