Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of D3×AGL(1,𝔽7) |
| automorphism | Aut(G) = C22×D3×AGL(1,𝔽7) (order 1008) · Inn(G) = D3×AGL(1,𝔽7) · Out(G) = C22 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| Fitting, socle | C2×C3×C7 ⋊ C22×C3 |
| derived | C3×C7 ⋊ C23×C3 |
| Hall | C3×ΓL(1,𝔽8) ⋊ C23 |
| Sylow | C7 ⋊ C22×C3×D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D3×AGL(1,𝔽7) |
| commutator, cosocle, hypocenter | C3×C7 | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C7 | C22×C3 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C2 | |
138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 42 |
A minimal faithful permutation representation — 3 generators on 12 points:
(3,4,5)(7,10,11,8,9,12);(1,2)(4,5)(6,7,9,12,11,10,8);(1,2)42 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 162);