Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of C3⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C7×C3⋊Dic3 |
| unambiguous semidirect product | (C2×C3)⋊(C7×Dic3) |
| derived, Sylow | C32 ⋊ C2×C4×C7 |
| Hall | C32×C7 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C7 | C3⋊D3 |
| commutator, hypocenter | C32 | C2×C4×C7 |
| Frattini | C2 | C2×C7×C3⋊D3 |
| Fitting, socle, prime radical | C22×C32×C7 | C2 |
| cosocle | C2×C32 | C22×C7 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C22×C7 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 504 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2)(4,5)(7,8)(9,10,11,12)(13,14,15,16,17,18,19);(1,2)(3,4,5);(6,7,8)168 conjugacy classes — view the full character table.
| maximal subgroup of | Dic7×C3⋊Dic3, (C2×D7)⋊(C3⋊Dic3), #11759168, … |
| maximal normal subgroup of | Dic7×C3⋊Dic3, (C2×D7)⋊(C3⋊Dic3), #11759168, … |
| maximal quotient of | C2×C7×C3⋊(C3⋊C8), C7×#3748, C4×C7×C3⋊Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 146);