Finite Group DatabasebetaFinite Group Databasebeta| order | 8064 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| direct product | ΓL(1,𝔽8)×#73081 |
| Fitting | C2×C7×#120 ⋊ C3×D3 |
| Sylow | C7 ⋊ C3×#73081 |
1026 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 2 generators on 19 points:
(2,4)(6,12,10,9)(7,11)(14,15,16)(17,19,18);(1,2)(3,4)(5,6)(7,10,11,9,8,12)(13,14,15,17,16,18,19)110 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((2,4)(6,12,10,9)(7,11)(14,15,16)(17,19,18), (1,2)(3,4)(5,6)(7,10,11,9,8,12)(13,14,15,17,16,18,19));