Finite Group DatabasebetaFinite Group Databasebeta| order | 11424 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C16 · Schur multiplier (H2) = C2 |
| direct product | AGL(1,𝔽7)×AGL(1,𝔽17) |
| Fitting, socle, derived, Hall | C7×C17 ⋊ C2×C3×C16 |
| Hall | D7×AGL(1,𝔽17) ⋊ C3 |
| Hall | C17×ΓL(1,𝔽8) ⋊ C2×C16 |
| Hall | ΓL(1,𝔽8) ⋊ C2×AGL(1,𝔽17) |
| Sylow | C17 ⋊ C16×AGL(1,𝔽7) |
| Sylow | C7 ⋊ C2×C3×AGL(1,𝔽17) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C7×C17 | C2×C3×C16 |
| prime radical | D17×AGL(1,𝔽7) | C8 |
| cosocle | C7×#3633 | C22×C3 |
| LCS | G ▷ C7×C17 | |
| UCS | C1 | |
112 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 119 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,3,4,8,5,9,7,14,6,10,11,16,12,13,15,17)(18,19,24)(21,22,23);(1,2,7,9,12,10,16,14,4,11,17,8,3,5,13,15,6)(19,20)(21,22)(23,24)119 conjugacy classes — view the full character table.
| maximal subgroup of | C2×AGL(1,𝔽7)×AGL(1,𝔽17) |
| maximal normal subgroup of | C2×AGL(1,𝔽7)×AGL(1,𝔽17) |
| maximal quotient of | C2×AGL(1,𝔽7)×AGL(1,𝔽17) |
| automorphism group of | D119 |
| holomorph of | C7×C17 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3,4,8,5,9,7,14,6,10,11,16,12,13,15,17)(18,19,24)(21,22,23), (1,2,7,9,12,10,16,14,4,11,17,8,3,5,13,15,6)(19,20)(21,22)(23,24));