Finite Group DatabasebetaFinite Group Databasebeta| order | 10206 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| direct product | C3×ΓL(1,𝔽8)×#4183 |
| Fitting | C3×C7×AΣL(1,𝔽27) ⋊ C2×C3 |
| derived | C7×AΣL(1,𝔽27) ⋊ C2×C32 |
| Hall | C3×ΓL(1,𝔽8)×AΣL(1,𝔽27) ⋊ C2 |
| Sylow | C7 ⋊ C32×#4183 |
2128 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 2128 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 126 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3,5,4,6,7)(8,9,10)(12,19,18,15,16,17)(13,14);(2,3,4)(5,7,6)(11,16,15,14,18,17,13,12,19);(12,16,18)(15,17,19)195 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#13091120, C3×AGL(1,𝔽7)×#4183 |
| maximal normal subgroup of | C3×AGL(1,𝔽7)×#4183 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3,5,4,6,7)(8,9,10)(12,19,18,15,16,17)(13,14), (2,3,4)(5,7,6)(11,16,15,14,18,17,13,12,19), (12,16,18)(15,17,19));