Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | ΓL(1,𝔽8)×#3025 |
| Fitting, Hall | C7×#3025 ⋊ C3 |
| Sylow | C7 ⋊ C3×#3025 |
1532 conjugacy classes of subgroups; only the 318 normal subgroups are shown.
This group has 1532 conjugacy classes of subgroups — too many to draw. Only its 318 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 — 4 generators on 23 points:
(2,3,4)(5,7,6)(8,11,12,19)(9,14,15,23)(10,16,17,18)(13,21,20,22);(1,2,4,6,3,5,7)(13,20)(21,22);(8,10)(9,13)(11,18)(12,17)(14,22)(15,20)(16,19)(21,23);(8,9)(10,13)(11,14)(12,15)(16,21)(17,20)(18,22)(19,23)160 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#3025 |
| maximal normal subgroup of | AGL(1,𝔽7)×#3025 |
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)(8,11,12,19)(9,14,15,23)(10,16,17,18)(13,21,20,22), (1,2,4,6,3,5,7)(13,20)(21,22), (8,10)(9,13)(11,18)(12,17)(14,22)(15,20)(16,19)(21,23), (8,9)(10,13)(11,14)(12,15)(16,21)(17,20)(18,22)(19,23));