Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) = C23×C3×D32 (order 864) · Inn(G) = D32 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | D3×C7×Dic3 |
| unambiguous semidirect product | (C3×D3)⋊(C4×C7) |
| derived, Sylow | C32 ⋊ C2×C4×C7 |
| Hall | C32×C7 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | D32 |
| commutator, hypocenter | C32 | C2×C4×C7 |
| Frattini | C2 | D32×C7 |
| Fitting, socle | C2×C32×C7 | C22 |
| prime radical | C2×C3×D3×C7 | C2 |
| cosocle | C2×C32 | C22×C7 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C7 | |
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)(7,10,9,8)(11,12,13,14,15,16,17);(2,3)(4,5,6)126 conjugacy classes — view the full character table.
| maximal subgroup of | D3×Dic3×D7, D3×D21⋊C4, D3×#4246, … |
| maximal normal subgroup of | D3×Dic3×D7, D3×D21⋊C4, D3×#4246, … |
| maximal quotient of | D3×C7×C3⋊C8, C7×#3711, C7×Dic32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 106);