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×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | A4⋊AGL(1,𝔽7) |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊S4 |
| unambiguous semidirect product | (C7⋊S4)⋊C3 |
| Fitting, socle | C22×C7 ⋊ C3×D3 |
| derived | C7×A4 ⋊ C2×C3 |
| Sylow | C7 ⋊ C3×S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C7×A4 | C2×C3 |
| Fitting, socle | C22×C7 | C3×D3 |
| LCS | G ▷ C7×A4 | |
| UCS | C1 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 28 |
A minimal faithful permutation representation — 2 generators on 11 points:
(1,2,4,3)(6,9,10,7,8,11);(2,3,4)(5,6,8,11,10,9,7)Polycyclic presentation:
2,3^2,7,2^2|c^a=c^2,d^a=d^6,e^a=f,f^a=e,d^b=d^2,e^c=f,f^c=ef19 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 160);