Finite Group DatabasebetaFinite Group Databasebeta| order | 5292 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2 |
| direct product | D7×#4430⋊C2 |
| Hall | C7×#4430 ⋊ C22 |
| Hall | D72 ⋊ 3−1+2 |
| Sylow | C72 ⋊ C22×3−1+2 |
172 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 126 |
A minimal faithful permutation representation — 2 generators on 23 points:
(2,5,6)(4,8,9)(11,12)(13,14)(15,16)(17,18)(20,21)(22,23);(1,6,9,3,5,4,7,2,8)(10,14,15,11,12,16,13)(18,23,21,20,19,22)155 conjugacy classes — view the full character table.
| maximal subgroup of | #72229823 |
| maximal normal subgroup of | #72229823 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,5,6)(4,8,9)(11,12)(13,14)(15,16)(17,18)(20,21)(22,23), (1,6,9,3,5,4,7,2,8)(10,14,15,11,12,16,13)(18,23,21,20,19,22));