Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of C3⋊D21 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊(C3⋊(C7⋊C8)) |
| unambiguous semidirect product | C7⋊(C3⋊(C3⋊C8)) |
| derived, Hall | C32×C7 ⋊ C8 |
| Sylow | C32 ⋊ C7⋊C8 |
| Sylow | C7 ⋊ C3⋊(C3⋊C8) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | C3⋊D21 |
| commutator, hypocenter | C32×C7 | C8 |
| Fitting, cosocle | C32×C4×C7 | C2 |
| socle, prime radical | C2×C32×C7 | C4 |
| LCS | G ▷ C32×C7 | |
| UCS | C1 ◁ C4 | |
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:
(2,3)(4,5)(6,7)(8,9,10,12,11,13,14,15)(17,18)(20,21);(1,2,4,6,7,5,3)(19,20,21);(16,17,18)Polycyclic presentation:
2^3,3^2,7|a^2=b,b^2=c,d^a=d^2,e^a=e^2,f^a=f^6132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 50);