Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of D3×D21 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| direct product | D3×Dic21 |
| unambiguous semidirect product | (C3×D3)⋊Dic7 |
| unambiguous semidirect product | (D3×C7)⋊Dic3 |
| derived, Hall | C32×C7 ⋊ C2×C4 |
| Sylow | C32 ⋊ C2×Dic7 |
| Sylow | C7 ⋊ D3×Dic3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×D21 |
| commutator, hypocenter | C32×C7 | C2×C4 |
| Fitting, socle, cosocle | C2×C32×C7 | C22 |
| prime radical | C2×C3×D3×C7 | C2 |
| LCS | G ▷ C32×C7 | |
| UCS | C1 ◁ C2 | |
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)(5,6)(7,8,9,10)(12,13)(14,15)(16,17);(2,3)(4,5,6)(11,12,14,16,17,15,13)72 conjugacy classes — view the full character table.
| maximal subgroup of | D3×Dic3×D7, D32×Dic7, D3×#4245, … |
| maximal normal subgroup of | D3×Dic3×D7, D32×Dic7, D3×#4245, … |
| maximal quotient of | D3×C3⋊(C7⋊C8), #11759192, Dic3×Dic21, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 112);