Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of D3×ΓL(1,𝔽8) |
| automorphism | Aut(G) = C22×D3×AGL(1,𝔽7) (order 1008) · Inn(G) = D3×ΓL(1,𝔽8) · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C4×D3×ΓL(1,𝔽8) |
| unambiguous semidirect product | (C4×D3×C7)⋊C3 |
| Fitting | C3×C4×C7 ⋊ C2×C3 |
| derived | C3×C7 ⋊ C2×C3×C4 |
| Hall | C3×ΓL(1,𝔽8) ⋊ C2×C4 |
| Sylow | C7 ⋊ C3×C4×D3 |
70 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)(9,10,11)(12,14,13);(1,2,3)(4,5,6,7)(8,9,10,12,11,13,14)60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 81);