Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | Aut(G) = C22×D3×AGL(1,𝔽7) (order 1008) · Inn(G) = AGL(1,𝔽7) · Out(G) = C22×D3 |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×C4×AGL(1,𝔽7) |
| unambiguous semidirect product | D7⋊(C32×C4) |
| unambiguous semidirect product | (C4×D7)⋊C32 |
| unambiguous semidirect product | (C3×C4×D7)⋊C3 |
| Fitting | C3×C4×C7 ⋊ C2×C3 |
| derived, Sylow | C7 ⋊ C2×C32×C4 |
| Hall | C3×ΓL(1,𝔽8) ⋊ C2×C4 |
| Hall | C4×D7 ⋊ C32 |
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:
(9,12,13,10,11,14);(1,2,3)(4,7,6,5)(8,9,11,14,13,12,10)84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 76);