Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25×C4 |
| direct product | ΓL(1,𝔽8)×#2679 |
| Fitting, Hall | C7×#2679 ⋊ C3 |
| Sylow | C7 ⋊ C3×#2679 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C23×C7 | C24×C3 |
| Fitting | C7×#2679 | C3 |
| hypercenter | #2679 | ΓL(1,𝔽8) |
| hypocenter | C7 | C3×#2679 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #2679 | |
2000 conjugacy classes of subgroups; only the 336 normal subgroups are shown.
This group has 2000 conjugacy classes of subgroups — too many to draw. Only its 336 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 19 points:
(2,4)(9,10)(11,12)(14,15,16)(17,19,18);(6,8)(9,10)(11,12)(13,14,16,18,15,17,19);(5,6)(7,8)(10,12);(1,2)(3,4)(10,12)160 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#2679, ΓL(1,𝔽8)×#13343, #72277381 |
| maximal normal subgroup of | AGL(1,𝔽7)×#2679, ΓL(1,𝔽8)×#13343, #72277381 |
| maximal quotient of | ΓL(1,𝔽8)×#15947 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,4)(9,10)(11,12)(14,15,16)(17,19,18), (6,8)(9,10)(11,12)(13,14,16,18,15,17,19), (5,6)(7,8)(10,12), (1,2)(3,4)(10,12));