Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of D7×D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×D7×D9 |
| unambiguous semidirect product | (C7×D9)⋊C22 |
| unambiguous semidirect product | (C9×D7)⋊C22 |
| unambiguous semidirect product | D63⋊C22 |
| Fitting | C2×C7×C9 ⋊ C22 |
| derived, Hall | C7×C9 ⋊ C23 |
| Sylow | C9 ⋊ C22×D7 |
| Sylow | C7 ⋊ C22×D9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D7×D9 |
| commutator, cosocle, hypocenter | C7×C9 | C23 |
| Frattini | C3 | C2×D3×D7 |
| Fitting | C2×C7×C9 | C22 |
| socle | C2×C3×C7 | C2×D3 |
| LCS | G ▷ C7×C9 | |
| UCS | C1 ◁ C2 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 126 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2)(3,4,6,8,9,7,5)(11,13)(12,16)(14,17)(15,18);(4,5)(6,7)(8,9)(10,11,14,12,15,18,16,17,13);(1,2)60 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D7×D9, D7×D36, D9×D28, … |
| maximal normal subgroup of | C4×D7×D9, D7×D36, D9×D28, … |
| maximal quotient of | D9×Dic14, #11758976, #11758977, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 59);