Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of Dic42 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×D84 |
| unambiguous semidirect product | C4⋊(C3×D21) |
| Fitting | C32×C4×C7 ⋊ C2 |
| Hall | C32×C7 ⋊ D4 |
| Sylow | C32 ⋊ D28 |
| Sylow | C7 ⋊ C3×D12 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C2×D21 |
| commutator, cosocle | C2×C3×C7 | C22×C3 |
| Frattini | C2 | C2×C3×D21 |
| Fitting | C32×C4×C7 | C2 |
| socle | C2×C32×C7 | C22 |
| hypercenter | C3×C4 | D21 |
| hypocenter | C3×C7 | C3×D4 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,3)(4,5,6)(7,10,9,8)(11,12,14,16,17,15,13);(5,6)(8,10)(12,13)(14,15)(16,17)135 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#66561, C3×#66562, C3×#66566, … |
| maximal normal subgroup of | C3×#66561, C3×#66562, C3×#66566, … |
| maximal quotient of | C3×#66623, C3×D168, C3×Dic84, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 140);