Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×D7×#997 |
| Fitting | C2×C7×H(5,𝔽2) ⋊ C2×C3 |
| derived | C7×H(5,𝔽2) ⋊ C22×C3 |
| Hall | C2×D7×H(5,𝔽2) ⋊ C3 |
| Sylow | C7 ⋊ C22×#997 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D7×C22⋊A4 |
| commutator, cosocle, hypocenter | C7×H(5,𝔽2) | C22×C3 |
| Frattini | C2 | C2×D7×C22⋊A4 |
| Fitting | C2×C7×H(5,𝔽2) | C2×C3 |
| socle | C22×C7 | C2×C22⋊A4 |
| LCS | G ▷ C7×H(5,𝔽2) | |
| UCS | C1 ◁ C22 | |
2450 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 2450 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2)(4,7,5)(8,9,10)(11,12,14,16,17,15,13);(4,8)(5,10)(12,13)(14,15)(16,17);(3,9,6,7)(4,5,8,10)110 conjugacy classes — view the full character table.
| maximal subgroup of | C2×AGL(1,𝔽7)×#997, C2×D7×#5935, D3×D7×#997, … |
| maximal normal subgroup of | C2×AGL(1,𝔽7)×#997, C2×D7×#5935, C2×D7×#5934 |
| maximal quotient of | D3×D7×#997 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(4,7,5)(8,9,10)(11,12,14,16,17,15,13), (4,8)(5,10)(12,13)(14,15)(16,17), (3,9,6,7)(4,5,8,10));