Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of A4 |
| automorphism | Aut(G) = C2×C32×S4 (order 432) · Inn(G) = A4 · Out(G) = C22×C32 |
| homology | Abelianization (H1) = C2×C7×C9 · Schur multiplier (H2) = C2 |
| direct product | C2×C7×C22⋊C9 |
| derived | C22 ⋊ C2×C7×C9 |
| Hall | C23×C7 ⋊ C9 |
| Sylow | C23 ⋊ C7×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C7 | A4 |
| commutator, hypocenter | C22 | C2×C7×C9 |
| Frattini | C3 | C2×C7×A4 |
| Fitting, socle, prime radical | C23×C3×C7 | C3 |
| cosocle | C22×C3 | C2×C3×C7 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C2×C3×C7 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 126 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2)(4,5,6)(7,8,9,10,11,12,13)(14,15,17,16,18,20,19,21,22);(3,6)(4,5)168 conjugacy classes — view the full character table.
| maximal subgroup of | C7×(C22⋊C9)⋊C4, C7⋊((C22⋊C9)⋊C4), Dic7×C22⋊C9, … |
| maximal normal subgroup of | C7×(C22⋊C9)⋊C4, C7⋊((C22⋊C9)⋊C4), Dic7×C22⋊C9, … |
| maximal quotient of | C4×C7×C22⋊C9, C2×C7×Q8⋊C9, C7×#41⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 61);