Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of SL(2,𝔽3) |
| automorphism | Aut(G) = C2×C32×S4 (order 432) · Inn(G) = A4 · Out(G) = C22×C32 |
| homology | Abelianization (H1) = C7×C9 · Schur multiplier (H2) = C1 |
| direct product | C7×Q8⋊C9 |
| unambiguous semidirect product | Q8⋊(C7×C9) |
| derived, Sylow | Q8 ⋊ C7×C9 |
| Hall | C7×Q8 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical, hypercenter | C2×C3×C7 | A4 |
| commutator, hypocenter | Q8 | C7×C9 |
| Frattini | C2×C3 | C7×A4 |
| Fitting | C3×C7×Q8 | C3 |
| cosocle | C3×Q8 | C3×C7 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C3×C7 | |
22 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 504 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,4,5,6,7)(9,10,12)(13,15,14)(16,23,22,18,17,24,21,20,19);(8,12,11,14)(9,15,13,10)147 conjugacy classes — view the full character table.
| derived subgroup of | C7⋊#3688, C7⋊#3689 |
| maximal subgroup of | C7×#3688, C7×#3689, C7⋊#3688, … |
| maximal normal subgroup of | C7×#3688, C7×#3689, C7⋊#3688, … |
| maximal quotient of | C2×C7×Q8⋊C9, C7×Q8⋊C27, C3×C7×Q8⋊C9 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 22);