Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#89331⋊C3 |
| Hall | C2×#89331 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C2×AΣL(1,𝔽16) |
| Sylow | C7 ⋊ C2×C3×AΣL(1,𝔽16) |
876 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 876 conjugacy classes of subgroups — too many to draw. Only its 121 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) = 112 |
A minimal faithful permutation representation — 3 generators on 17 points:
(3,4,9)(5,6,8)(11,14)(15,16);(1,2);(4,5)(6,8)(7,9)(10,11,12,15,13,16,17,14)106 conjugacy classes — view the full character table.
| maximal subgroup of | C2×AGL(1,𝔽7)×AΣL(1,𝔽16), #72276599, #72276646, … |
| maximal normal subgroup of | C2×AGL(1,𝔽7)×AΣL(1,𝔽16), #72276599, #72276646, … |
| maximal quotient of | #72276646 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4,9)(5,6,8)(11,14)(15,16), (1,2), (4,5)(6,8)(7,9)(10,11,12,15,13,16,17,14));