Finite Group DatabasebetaFinite Group Databasebeta| order | 16128 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | AGL(1,𝔽7)×#3532⋊C3 |
| Fitting | C7×#3532 ⋊ C2×C32 |
| derived | C7×H(3,𝔽4) ⋊ C22×C32 |
| Hall | D7×#3532 ⋊ C32 |
| Sylow | C7 ⋊ C2×C3×#3532⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C7×H(3,𝔽4) | C22×C32 |
| Frattini | C22 | C2×AGL(1,𝔽7)×C22⋊A4 |
| Fitting | C7×#3532 | C2×C32 |
| socle | C22×C7 | C22×C3×C22⋊A4 |
| LCS | G ▷ C7×H(3,𝔽4) | |
| UCS | C1 | |
7044 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 7044 conjugacy classes of subgroups — too many to draw. Only its 111 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:
(1,2,5,6,4,7,3)(9,20,18,13,14,17)(10,22,21,16,15,23)(11,12,19);(2,6,5,3,4,7)(9,22)(10,16)(13,15)(14,21)(17,18)(20,23);(8,20,19,14)(9,16,22,17)(10,13,18,15)(11,21,12,23)126 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,5,6,4,7,3)(9,20,18,13,14,17)(10,22,21,16,15,23)(11,12,19), (2,6,5,3,4,7)(9,22)(10,16)(13,15)(14,21)(17,18)(20,23), (8,20,19,14)(9,16,22,17)(10,13,18,15)(11,21,12,23));