Finite Group DatabasebetaFinite Group Databasebeta| order | 5376 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C25 |
| direct product | D7×#3532⋊C3 |
| Fitting | C7×#3532 ⋊ C2×C3 |
| derived | C7×H(3,𝔽4) ⋊ C22×C3 |
| Hall | D7×#3532 ⋊ C3 |
| Sylow | C7 ⋊ C2×#3532⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C7×H(3,𝔽4) | C22×C3 |
| Frattini | C22 | C2×D7×C22⋊A4 |
| Fitting | C7×#3532 | C2×C3 |
| socle | C22×C7 | C22×C22⋊A4 |
| LCS | G ▷ C7×H(3,𝔽4) | |
| UCS | C1 | |
3394 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 3394 conjugacy classes of subgroups — too many to draw. Only its 59 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:
(1,2,4,6,7,5,3)(9,20,17,13,14,18)(10,22,21,16,15,23)(11,19,12);(2,3)(4,5)(6,7)(9,22)(10,16)(13,15)(14,21)(17,18)(20,23);(8,20,19,14)(9,16,22,18)(10,13,17,15)(11,23,12,21)90 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#3532⋊C3 |
| maximal normal subgroup of | AGL(1,𝔽7)×#3532⋊C3 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,4,6,7,5,3)(9,20,17,13,14,18)(10,22,21,16,15,23)(11,19,12), (2,3)(4,5)(6,7)(9,22)(10,16)(13,15)(14,21)(17,18)(20,23), (8,20,19,14)(9,16,22,18)(10,13,17,15)(11,23,12,21));