Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of D3×D21 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived, Hall | C32×C7 ⋊ C2×C4 |
| Sylow | C32 ⋊ C4×D7 |
| Sylow | C7 ⋊ #624 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×D21 |
| commutator, hypocenter | C32×C7 | C2×C4 |
| Fitting, socle, cosocle | C2×C32×C7 | C22 |
| prime radical | C2×C3⋊D21 | C2 |
| LCS | G ▷ C32×C7 | |
| UCS | C1 ◁ C2 | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 2 generators on 17 points:
(2,3)(4,5,6)(7,9,10,8)(11,12,14,16,17,15,13);(1,2,3)(5,6)(7,8,10,9)(12,13)(14,15)(16,17)Polycyclic presentation:
2^3,3^2,7|a^2=c,b^2=c,e^a=e^2,f^a=f^6,d^b=d^272 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#624, D3×D21⋊C4, #11759427, … |
| maximal normal subgroup of | D7×#624, D3×D21⋊C4, #11759427, … |
| maximal quotient of | #11759190, #11759193, Dic3×Dic21, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 113);