Finite Group DatabasebetaFinite Group Databasebeta| order | 8064 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C2×AGL(1,𝔽7)×#120⋊C3 |
| Fitting | C2×C7×#120 ⋊ C2×C32 |
| derived | C24×C7 ⋊ C23×C32 |
| Hall | C2×D7×#120 ⋊ C32 |
| Sylow | C7 ⋊ C22×C3×#120⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | AGL(1,𝔽7)×#120⋊C3 |
| commutator, cosocle, hypocenter | C24×C7 | C23×C32 |
| Frattini | C22 | C22×A4×AGL(1,𝔽7) |
| Fitting | C2×C7×#120 | C2×C32 |
| socle | C23×C7 | C22×C3×A4 |
| LCS | G ▷ C24×C7 | |
| UCS | C1 ◁ C2 | |
3648 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 3648 conjugacy classes of subgroups — too many to draw. Only its 155 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2)(5,9,6)(7,10,8)(11,12,14,17,16,15,13);(4,10)(7,8)(12,15,16,13,14,17);(3,4)(5,7)(6,8)(9,10)140 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(5,9,6)(7,10,8)(11,12,14,17,16,15,13), (4,10)(7,8)(12,15,16,13,14,17), (3,4)(5,7)(6,8)(9,10));