Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of D4×ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×D4×ΓL(1,𝔽8) |
| unambiguous semidirect product | C4⋊(C2×C3×ΓL(1,𝔽8)) |
| unambiguous semidirect product | (C7×D4)⋊C32 |
| unambiguous semidirect product | (C3×C7×D4)⋊C3 |
| Fitting | C3×C7×D4 ⋊ C3 |
| Hall | C7×D4 ⋊ C32 |
| Sylow | C7 ⋊ C32×D4 |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 2 generators on 14 points:
(2,3,4)(5,7,6)(9,11);(1,2,3,5,4,6,7)(8,9)(10,11)(12,13,14)75 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 124);