Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
| direct product | C2×C7⋊#5935 |
| Fitting | C2×C7×H(5,𝔽2) ⋊ D3 |
| derived | C7×#997 ⋊ C22 |
| Sylow | C7 ⋊ C2×#5935 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C7⋊#1020 |
| commutator, cosocle, hypocenter | C7×#997 | C22 |
| Frattini | C2 | C2×C7⋊#1020 |
| Fitting | C2×C7×H(5,𝔽2) | D3 |
| socle | C22×C7 | #1020 |
| LCS | G ▷ C7×#997 | |
| UCS | C1 ◁ C22 | |
666 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 666 conjugacy classes of subgroups — too many to draw. Only its 33 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 — 2 generators on 17 points:
(1,2)(4,5,8)(7,10,9)(11,12,14,16,17,15,13);(3,8,4,5,6,9,7,10)(12,13)(14,15)(16,17)92 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72214048, C2×D7×#5935 |
| maximal normal subgroup of | C2×#72214048, C2×D7×#5935 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(4,5,8)(7,10,9)(11,12,14,16,17,15,13), (3,8,4,5,6,9,7,10)(12,13)(14,15)(16,17));