Finite Group DatabasebetaFinite Group Databasebeta| order | 5376 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#11710731 |
| Fitting, Hall | C7×#62817 ⋊ C3 |
| derived | H(3,𝔽4) ⋊ C22×C3×C7 |
| Sylow | #62817 ⋊ C3×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | #11710731 |
| commutator, cosocle, hypocenter | H(3,𝔽4) | C22×C3×C7 |
| Frattini | C22 | C22×C7×C22⋊A4 |
| Fitting | C7×#62817 | C3 |
| socle | C22×C7 | C22×C22⋊A4 |
| LCS | G ▷ H(3,𝔽4) | |
| UCS | C1 ◁ C7 | |
3368 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 3368 conjugacy classes of subgroups — too many to draw. Only its 82 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 — 3 generators on 23 points:
(1,2,3,4,5,6,7)(9,20,23,14,15,22)(10,21,13,18,16,17)(11,19,12);(9,17)(10,18)(13,14)(15,20);(8,21,12,16)(9,18,13,15)(10,14,20,17)(11,23,19,22)252 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#11710731 |
| maximal normal subgroup of | ΓL(1,𝔽8)×#11710731 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3,4,5,6,7)(9,20,23,14,15,22)(10,21,13,18,16,17)(11,19,12), (9,17)(10,18)(13,14)(15,20), (8,21,12,16)(9,18,13,15)(10,14,20,17)(11,23,19,22));