Finite Group DatabasebetaFinite Group Databasebeta| order | 2240 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C2×AGL(1,𝔽5)×AGL(1,𝔽8) |
| Fitting, socle | C24×C5 ⋊ C4×C7 |
| derived | C23×C5 ⋊ C2×C4×C7 |
| Hall | C24×AGL(1,𝔽5) ⋊ C7 |
| Sylow | C5 ⋊ C2×C4×AGL(1,𝔽8) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | AGL(1,𝔽5)×AGL(1,𝔽8) |
| commutator, hypocenter | C23×C5 | C2×C4×C7 |
| Frattini | C1 | G |
| Fitting, socle | C24×C5 | C4×C7 |
| prime radical | C2×D5×AGL(1,𝔽8) | C2 |
| cosocle | C23×D5 | C22×C7 |
| LCS | G ▷ C23×C5 | |
| UCS | C1 ◁ C2 | |
254 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 254 conjugacy classes of subgroups — too many to draw. Only its 30 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) = 70 |
A minimal faithful permutation representation — 2 generators on 15 points:
(4,7,10,5,6,8,9)(12,13,14,15);(1,2)(3,4)(5,7)(6,8)(9,10)(11,12,15,13,14)80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×AGL(1,𝔽5)×AΓL(1,𝔽8) |
| maximal normal subgroup of | C2×AGL(1,𝔽5)×AΓL(1,𝔽8) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,7,10,5,6,8,9)(12,13,14,15), (1,2)(3,4)(5,7)(6,8)(9,10)(11,12,15,13,14));