Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of #92688 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Hall | C32×C7 ⋊ Q8 |
| Sylow | C32 ⋊ Dic14 |
| Sylow | C7 ⋊ #627 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×D21 |
| commutator, Fitting, socle, prime radical, cosocle | C2×C32×C7 | C22 |
| hypocenter | C32×C7 | Q8 |
| LCS | G ▷ C2×C32×C7 ▷ C32×C7 | |
| UCS | C1 ◁ C2 | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 2 generators on 21 points:
(2,3)(4,5,6)(7,8,10,12,13,11,9)(14,16,17,21)(15,18,19,20);(1,2,3)(5,6)(8,9)(10,11)(12,13)(14,15,17,19)(16,20,21,18)Polycyclic presentation:
2^3,3^2,7|a^2=c,b^2=c,b^a=bc,e^a=e^2,f^a=f^6,d^b=d^269 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#627, D3×#4248, #11759435, … |
| maximal normal subgroup of | D7×#627, D3×#4248, #11759435, … |
| maximal quotient of | #11759210, #11759211, Dic3⋊Dic21, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 117);