Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C25 |
| direct product | ΓL(1,𝔽8)×#2021 |
| Fitting, Hall | C7×#2021 ⋊ C3 |
| Sylow | C7 ⋊ C3×#2021 |
1276 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1276 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 112 |
A minimal faithful permutation representation — 3 generators on 19 points:
(6,7,8)(9,11,10)(12,14)(13,16,17,19)(15,18);(1,2,3,4)(5,6,7,9,8,10,11)(12,13,15,17)(14,16)(18,19);(2,4)(16,19)130 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#2021, #72276695, ΓL(1,𝔽8)×#23900, … |
| maximal normal subgroup of | AGL(1,𝔽7)×#2021, #72276695, ΓL(1,𝔽8)×#23900 |
| maximal quotient of | ΓL(1,𝔽8)×#13343, ΓL(1,𝔽8)×#10336 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((6,7,8)(9,11,10)(12,14)(13,16,17,19)(15,18), (1,2,3,4)(5,6,7,9,8,10,11)(12,13,15,17)(14,16)(18,19), (2,4)(16,19));