Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of #92676 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Hall | C32×C7 ⋊ Q8 |
| Sylow | C32 ⋊ Dic14 |
| Sylow | C7 ⋊ #634 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D7×C3⋊D3 |
| commutator, Fitting, socle, prime radical, cosocle | C2×C32×C7 | C22 |
| hypocenter | C32×C7 | Q8 |
| LCS | G ▷ C2×C32×C7 ▷ C32×C7 | |
| UCS | C1 ◁ C2 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 504 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,4,6,7,5,3)(8,10,11,15)(9,12,13,14)(17,18)(20,21);(2,3)(4,5)(6,7)(8,9,11,13)(10,14,15,12)(19,20,21);(16,17,18)Polycyclic presentation:
2^3,3^2,7|a^2=c,b^2=c,b^a=bc,f^a=f^6,d^b=d^2,e^b=e^257 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#4248, #11759435, #11759447, … |
| maximal normal subgroup of | D3×#4248, #11759435, #11759447, … |
| maximal quotient of | Dic7⋊(C3⋊Dic3), #11759170, #11759171, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 105);