Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of #92676 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Hall | C32×C7 ⋊ D4 |
| Sylow | C32 ⋊ C22⋊D7 |
| Sylow | C7 ⋊ #638 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D7×C3⋊D3 |
| commutator, Fitting, socle, cosocle | C2×C32×C7 | C22 |
| hypocenter | C32×C7 | D4 |
| LCS | G ▷ C2×C32×C7 ▷ C32×C7 | |
| UCS | C1 ◁ C2 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 252 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,4,6,7,5,3)(9,11)(13,14)(16,17);(2,3)(4,5)(6,7)(8,9)(10,11)(15,16,17);(12,13,14)Polycyclic presentation:
2^3,3^2,7|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×#4245, #11759425, #11759439, … |
| maximal normal subgroup of | D3×#4245, #11759425, #11759439, … |
| maximal quotient of | #11759153, #11759156, #11759157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 102);