Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C9 · Schur multiplier (H2) = C2 |
| direct product | C2×C8×C7⋊C9 |
| unambiguous semidirect product | (C2×C7×C8)⋊C9 |
| derived, Sylow | C7 ⋊ C2×C8×C9 |
| Hall | C2×C7×C8 ⋊ C9 |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 1008 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2)(12,13,14)(15,17,16)(18,19,21,20,22,24,23,25,26);(3,4,5,6,7,8,9,10)(11,12,13,15,14,16,17)240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 51);