Finite Group DatabasebetaFinite Group Databasebeta| order | 8064 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×#72214620 |
| Fitting | C2×C7×#120 ⋊ C3×D3 |
| derived | C7×C22⋊A4 ⋊ C23×C3 |
| Sylow | C7 ⋊ C2×C3×#1020⋊C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72214620 |
| commutator, cosocle, hypocenter | C7×C22⋊A4 | C23×C3 |
| Frattini | C22 | C22×A4⋊AGL(1,𝔽7) |
| Fitting | C2×C7×#120 | C3×D3 |
| socle | C23×C7 | C2×C3×S4 |
| LCS | G ▷ C7×C22⋊A4 | |
| UCS | C1 ◁ C2 | |
1890 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 1890 conjugacy classes of subgroups — too many to draw. Only its 73 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)(4,10,8)(5,6,9)(11,12,17)(14,15,16);(5,9)(7,10)(12,13)(14,15)(16,17);(3,4)(5,7)(6,8)(9,10)104 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(4,10,8)(5,6,9)(11,12,17)(14,15,16), (5,9)(7,10)(12,13)(14,15)(16,17), (3,4)(5,7)(6,8)(9,10));