Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | ΓL(1,𝔽8)×#2196 |
| Fitting, Hall | C7×#2196 ⋊ C3 |
| Sylow | C7 ⋊ C3×#2196 |
708 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 708 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 56 |
A minimal faithful permutation representation — 3 generators on 15 points:
(2,3,4)(5,7,6)(9,12,14,13);(1,2,3,5,4,6,7)(12,13);(8,9,11,14)(10,12,15,13)100 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#2196, ΓL(1,𝔽8)×#73026 |
| maximal normal subgroup of | AGL(1,𝔽7)×#2196 |
| maximal quotient of | ΓL(1,𝔽8)×#13343 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3,4)(5,7,6)(9,12,14,13), (1,2,3,5,4,6,7)(12,13), (8,9,11,14)(10,12,15,13));