Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of D63 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| direct product | C2×Dic63 |
| unambiguous semidirect product | (C2×C7)⋊Dic9 |
| unambiguous semidirect product | (C2×C9)⋊Dic7 |
| derived, Hall | C7×C9 ⋊ C2×C4 |
| Sylow | C9 ⋊ C2×Dic7 |
| Sylow | C7 ⋊ C2×Dic9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D63 |
| commutator, hypocenter | C7×C9 | C2×C4 |
| Frattini | C2×C3 | C2×D21 |
| Fitting | C22×C7×C9 | C2 |
| socle, prime radical | C22×C3×C7 | D3 |
| cosocle | C2×C7×C9 | C22 |
| LCS | G ▷ C7×C9 | |
| UCS | C1 ◁ C22 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 504 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2)(3,4,5,6)(8,9)(10,11)(12,13)(15,17)(16,20)(18,19)(21,22);(1,2)(7,8,10,12,13,11,9)(14,22,17,20,19,18,16,15,21)132 conjugacy classes — view the full character table.
| maximal subgroup of | Dic7×Dic9, #11758871, Dic9⋊Dic7, … |
| maximal normal subgroup of | Dic7×Dic9, #11758871, Dic9⋊Dic7, … |
| maximal quotient of | C2×C7⋊(C9⋊C8), #11758925, C4×Dic63, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 37);