Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | AGL(1,𝔽7)×AΣL(1,𝔽16) |
| Fitting | C7×AΣL(1,𝔽16) ⋊ C2×C3 |
| Hall | D7×AΣL(1,𝔽16) ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C2×AΣL(1,𝔽16) |
| Sylow | C7 ⋊ C2×C3×AΣL(1,𝔽16) |
876 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 876 conjugacy classes of subgroups — too many to draw. Only its 101 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:
(2,5,6,3,4,7);(1,2,4,7,6,5,3)(9,12)(13,15);(8,9,10,13)(11,15)(12,14)91 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((2,5,6,3,4,7), (1,2,4,7,6,5,3)(9,12)(13,15), (8,9,10,13)(11,15)(12,14));