Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of #4245⋊C3 |
| automorphism | Aut(G) = C22×D3×AGL(1,𝔽7) (order 1008) · Inn(G) = D3×AGL(1,𝔽7) · Out(G) = C22 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #4248⋊C3 |
| Hall | C3×ΓL(1,𝔽8) ⋊ Q8 |
| Sylow | C7 ⋊ C3×Dic6 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×AGL(1,𝔽7) |
| commutator, Fitting, socle, cosocle | C2×C3×C7 | C22×C3 |
| prime radical | C2×C3×ΓL(1,𝔽8) | C22 |
| hypocenter | C3×C7 | C3×Q8 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2 | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 2 generators on 18 points:
(2,3)(4,5,10)(6,7,9)(11,13,14,18)(12,15,16,17);(1,2,3)(5,6)(7,9)(8,10)(11,12,14,16)(13,17,18,15)Polycyclic presentation:
2^2,3,2,3,7|a^2=d,b^2=d,b^a=bd,f^a=f^6,e^b=e^2,f^c=f^433 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 73);