Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C24 |
| direct product | ΓL(1,𝔽8)×#1304 |
| Fitting, Hall | C7×#1304 ⋊ C3 |
| Sylow | C7 ⋊ C3×#1304 |
792 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 792 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 23 points:
(2,3,4)(5,7,6)(8,10)(9,12)(11,15)(13,14)(16,18)(17,20,21,22)(19,23);(1,2,3,5,4,6,7)(8,9)(10,14,15,12)(11,13)(16,17)(18,22)(19,21)(20,23)130 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#10336, AGL(1,𝔽7)×#1304, #72278080 |
| maximal normal subgroup of | ΓL(1,𝔽8)×#10336, AGL(1,𝔽7)×#1304, #72278080 |
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)(8,10)(9,12)(11,15)(13,14)(16,18)(17,20,21,22)(19,23), (1,2,3,5,4,6,7)(8,9)(10,14,15,12)(11,13)(16,17)(18,22)(19,21)(20,23));