Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | AGL(1,𝔽7)×#492 |
| Fitting | C7×#492 ⋊ C2×C3 |
| Hall | D7×#492 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C2×#492 |
| Sylow | C7 ⋊ C2×C3×#492 |
1312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,7)(3,4,6)(12,13,14,16,15,17,18,19);(8,9,10,11);(2,3)(4,6)(5,7);(9,11)(13,16)(14,18)(17,19)154 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((1,2,7)(3,4,6)(12,13,14,16,15,17,18,19), (8,9,10,11), (2,3)(4,6)(5,7), (9,11)(13,16)(14,18)(17,19));