Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×AΣL(1,𝔽64) |
| Fitting, Hall | C7×#2846 ⋊ C3 |
| Sylow | #2846 ⋊ C3×C7 |
1254 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 1254 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 2 generators on 19 points:
(2,4)(7,8,11)(9,10,12)(13,14,15,16,17,18,19);(1,2)(3,4)(5,12,11,6)(7,10,8,9)196 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×AΣL(1,𝔽64) |
| maximal normal subgroup of | ΓL(1,𝔽8)×AΣL(1,𝔽64) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,4)(7,8,11)(9,10,12)(13,14,15,16,17,18,19), (1,2)(3,4)(5,12,11,6)(7,10,8,9));