Finite Group DatabasebetaFinite Group Databasebeta| order | 4032 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | AGL(1,𝔽7)×#986 |
| Fitting | C7×#41 ⋊ C2×C3×D3 |
| derived | C7×SL(2,𝔽3) ⋊ C23×C3 |
| Sylow | C7 ⋊ C2×C3×#986 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×S4×AGL(1,𝔽7) |
| commutator, cosocle, hypocenter | C7×SL(2,𝔽3) | C23×C3 |
| Fitting | C7×#41 | C2×C3×D3 |
| socle | C2×C7 | C22×C3×S4 |
| hypercenter | C4 | S4×AGL(1,𝔽7) |
| LCS | G ▷ C7×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 | |
702 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 702 conjugacy classes of subgroups — too many to draw. Only its 68 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:
(1,8,9,2,12,13,5,14,3,6,4,7)(10,16,11,15)(17,18,23)(20,21,22);(18,19)(20,21)(22,23);(2,6)(3,9)(4,10)(8,16)(11,12)(14,15)91 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#5429 |
| maximal normal subgroup of | AGL(1,𝔽7)×#5429 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,8,9,2,12,13,5,14,3,6,4,7)(10,16,11,15)(17,18,23)(20,21,22), (18,19)(20,21)(22,23), (2,6)(3,9)(4,10)(8,16)(11,12)(14,15));