Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | ΓL(1,𝔽8)×#1790 |
| Fitting, Hall | C7×#1790 ⋊ C3 |
| Sylow | C7 ⋊ C3×#1790 |
1340 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 1340 conjugacy classes of subgroups — too many to draw. Only its 204 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) = 224 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3,4)(6,7,8)(9,11,10)(16,18);(5,6,8,10,7,9,11)(12,14)(13,16)(15,19)(17,18);(12,13,15,17)(14,18,19,16)160 conjugacy classes — view the full character table.
| maximal subgroup of | #72277311, AGL(1,𝔽7)×#1790, ΓL(1,𝔽8)×#32898 |
| maximal normal subgroup of | #72277311, AGL(1,𝔽7)×#1790, ΓL(1,𝔽8)×#32898 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3,4)(6,7,8)(9,11,10)(16,18), (5,6,8,10,7,9,11)(12,14)(13,16)(15,19)(17,18), (12,13,15,17)(14,18,19,16));