Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of A4×AGL(1,𝔽7) |
| automorphism | Aut(G) = S4×AGL(1,𝔽7) (order 1008) · Inn(G) = G · Out(G) = C2 |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2×C3 |
| Fitting, socle, derived | C22×C7 ⋊ C2×C32 |
| Hall | C22×D7 ⋊ C32 |
| Sylow | C7 ⋊ C2×C3×A4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C22×C7 | C2×C32 |
| LCS | G ▷ C22×C7 | |
| UCS | C1 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 28 |
A minimal faithful permutation representation — 2 generators on 11 points:
(2,5,6,3,4,7)(8,11)(9,10);(1,2,4,7,6,5,3)(9,10,11)28 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 161);