Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of C3⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C7×C8 · Schur multiplier (H2) = C3 |
| direct product | C7×C3⋊(C3⋊C8) |
| unambiguous semidirect product | C3⋊(C7×C3⋊C8) |
| derived, Sylow | C32 ⋊ C7×C8 |
| Hall | C32×C7 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C7 | C3⋊D3 |
| commutator, hypocenter | C32 | C7×C8 |
| Frattini | C4 | C7×C3⋊D3 |
| Fitting | C32×C4×C7 | C2 |
| socle, prime radical | C2×C32×C7 | C4 |
| cosocle | C32×C4 | C2×C7 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C4×C7 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 504 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,8,7,6,4,5,3,2)(10,11)(13,14)(15,16,17,18,19,20,21);(12,13,14);(9,10,11)168 conjugacy classes — view the full character table.
| maximal subgroup of | C7×#3708, C7⋊#3708, D7×C3⋊(C3⋊C8), … |
| maximal normal subgroup of | C7×#3708, C7⋊#3708, D7×C3⋊(C3⋊C8), … |
| maximal quotient of | C7×C3⋊(C3⋊C16), C2×C7×C3⋊(C3⋊C8), C7×C3⋊(C9⋊C8), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 49);