Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
| direct product | C2×#1020⋊D7 |
| Fitting | C25×C7 ⋊ C2×D3 |
| derived | C7×C22⋊A4 ⋊ C23 |
| Sylow | C7 ⋊ C2×#1020⋊C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #1020⋊D7 |
| commutator, cosocle, hypocenter | C7×C22⋊A4 | C23 |
| Frattini | C22 | C2×D7×S4 |
| Fitting | C25×C7 | C2×D3 |
| socle | C23×C7 | C2×S4 |
| LCS | G ▷ C7×C22⋊A4 | |
| UCS | C1 ◁ C2 | |
926 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 926 conjugacy classes of subgroups — too many to draw. Only its 39 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) = 84 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2)(4,8,7)(5,6,9)(11,12,14,16,17,15,13);(6,9)(8,10);(3,4)(5,7)(6,8)(9,10)(12,13)(14,15)(16,17)88 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72214629 |
| maximal normal subgroup of | C2×#72214629 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(4,8,7)(5,6,9)(11,12,14,16,17,15,13), (6,9)(8,10), (3,4)(5,7)(6,8)(9,10)(12,13)(14,15)(16,17));