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 | C4×C7×C3⋊D3 |
| Fitting | C32×C4×C7 ⋊ C2 |
| derived, Sylow | C32 ⋊ C2×C4×C7 |
| Hall | C32×C7 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C7 | C3⋊D3 |
| commutator, hypocenter | C32 | C2×C4×C7 |
| Frattini | C2 | C2×C7×C3⋊D3 |
| Fitting | C32×C4×C7 | C2 |
| socle | C2×C32×C7 | C22 |
| prime radical | C2×C7×C3⋊D3 | C2 |
| cosocle | C2×C32 | C22×C7 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C4×C7 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 252 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,3,4,5,6,7)(13,14)(16,17);(8,9,10,11)(15,16,17);(12,13,14)168 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×C7⋊C8, #11759151, C7×#3710, … |
| maximal normal subgroup of | C3⋊D3×C7⋊C8, #11759151, C7×#3710, … |
| maximal quotient of | C7×C8×C3⋊D3, C7×#3743, C4×C7×C3⋊Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 144);