Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of D21 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C1 |
| direct product | C3×C3⋊(C7⋊C8) |
| unambiguous semidirect product | C3⋊(C3×C7⋊C8) |
| derived | C3×C7 ⋊ C3×C8 |
| Hall | C32×C7 ⋊ C8 |
| Sylow | C32 ⋊ C7⋊C8 |
| Sylow | C7 ⋊ C3×C3⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | D21 |
| commutator, hypocenter | C3×C7 | C3×C8 |
| Frattini | C4 | C3×D21 |
| Fitting | C32×C4×C7 | C2 |
| socle, prime radical | C2×C32×C7 | C4 |
| cosocle | C3×C4×C7 | C2×C3 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C3×C4 | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3)(5,6)(8,9)(10,11)(12,13)(14,18,20,15,17,21,16,19);(4,5,6)(7,8,10,12,13,11,9)144 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D7×C3⋊C8, C3×D3×C7⋊C8, C3×#66557, … |
| maximal normal subgroup of | C3×D7×C3⋊C8, C3×D3×C7⋊C8, C3×#66557, … |
| maximal quotient of | C3×C3⋊(C7⋊C16), C2×C3×C3⋊(C7⋊C8), C3×C7⋊(C9⋊C8), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 48);