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) = C32×C8 · Schur multiplier (H2) = C3 |
| direct product | C3×ΓL(1,𝔽8)⋊C8 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊(C3×C8) |
| unambiguous semidirect product | (C7⋊C8)⋊C32 |
| unambiguous semidirect product | (C3×C7⋊C8)⋊C3 |
| derived, Sylow | C7 ⋊ C32×C8 |
| Hall | C3×ΓL(1,𝔽8) ⋊ C8 |
| Hall | C7⋊C8 ⋊ C32 |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 2 generators on 18 points:
(5,9,8,6,7,10)(11,15,17,12,14,18,13,16);(1,2,3)(4,5,8,9,7,10,6)84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 42);