Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C2×C8 |
| direct product | ΓL(1,𝔽8)×#1721 |
| Fitting, Hall | C7×#1721 ⋊ C3 |
| Sylow | C7 ⋊ C3×#1721 |
484 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 484 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 448 |
A minimal faithful permutation representation — 3 generators on 23 points:
(2,3,4)(5,7,6)(8,9)(10,12)(11,14)(13,15)(17,20)(18,22)(21,23);(1,2,4,6,3,5,7)(9,12)(10,15)(13,14)(17,21)(18,22)(20,23);(16,17,22,21,19,23,18,20)130 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#1721, ΓL(1,𝔽8)×#19745, #72259644 |
| maximal normal subgroup of | AGL(1,𝔽7)×#1721, ΓL(1,𝔽8)×#19745, #72259644 |
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)(8,9)(10,12)(11,14)(13,15)(17,20)(18,22)(21,23), (1,2,4,6,3,5,7)(9,12)(10,15)(13,14)(17,21)(18,22)(20,23), (16,17,22,21,19,23,18,20));