Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of (C3⋊D3)⋊D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived, Hall | C32×C7 ⋊ C2×C4 |
| Sylow | C32 ⋊ C2×Dic7 |
| Sylow | C7 ⋊ #624 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | (C3⋊D3)⋊D7 |
| commutator, hypocenter | C32×C7 | C2×C4 |
| Fitting, socle, cosocle | C2×C32×C7 | C22 |
| prime radical | C2×C7×C3⋊D3 | 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:
(1,2)(5,6)(11,12,14,16,17,15,13);(2,3)(4,5,6)(7,8,9,10)(12,13)(14,15)(16,17)Polycyclic presentation:
2^3,3^2,7|a^2=c,d^a=d^2,f^a=f^6,d^b=d^2,e^b=e^254 conjugacy classes — view the full character table.
| maximal subgroup of | D32⋊Dic7, #11759402, #11759407, … |
| maximal normal subgroup of | D32⋊Dic7, #11759402, #11759407, … |
| maximal quotient of | #11759214, #11759216, #11759224, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 119);