Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of #634 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C2×C3 |
| direct product | C7×C4⋊(C3⋊D3) |
| unambiguous semidirect product | C4⋊(C7×C3⋊D3) |
| Fitting | C32×C4×C7 ⋊ C2 |
| Hall | C32×C7 ⋊ D4 |
| Sylow | C32 ⋊ C7×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7 | C2×C3⋊D3 |
| commutator, cosocle | C2×C32 | C22×C7 |
| Frattini | C2 | C2×C7×C3⋊D3 |
| Fitting | C32×C4×C7 | C2 |
| socle | C2×C32×C7 | C22 |
| hypercenter | C4×C7 | C3⋊D3 |
| hypocenter | C32 | C7×D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C7 ◁ C4×C7 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 252 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,4,3,2)(8,9,10)(11,12,13,14,15,16,17);(2,4)(6,7)(9,10);(5,6,7)147 conjugacy classes — view the full character table.
| maximal subgroup of | #11759153, #11759155, #11759157, … |
| maximal normal subgroup of | #11759153, #11759155, #11759157, … |
| maximal quotient of | C7×#3744, C7×#3745, C7×#3746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 145);