Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of C3⋊AGL(1,𝔽7) |
| automorphism | Aut(G) = C22×D3×AGL(1,𝔽7) (order 1008) · Inn(G) = C3⋊AGL(1,𝔽7) · Out(G) = C23 |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(ΓL(1,𝔽8)⋊C8) |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊(C3⋊C8) |
| unambiguous semidirect product | (C3⋊(C7⋊C8))⋊C3 |
| derived | C3×C7 ⋊ C3×C8 |
| Hall | C3×ΓL(1,𝔽8) ⋊ C8 |
| Sylow | C7 ⋊ C3×C3⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | C3⋊AGL(1,𝔽7) |
| commutator, hypocenter | C3×C7 | C3×C8 |
| Fitting, cosocle | C3×C4×C7 | C2×C3 |
| socle | C2×C3×C7 | C3×C4 |
| prime radical | C2×C3×ΓL(1,𝔽8) | C4 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C4 | |
38 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)(5,8,9,6,7,10)(11,12,13,15,14,16,17,18);(1,2,3)(4,5,7,10,9,8,6)Polycyclic presentation:
2,3,2^2,3,7|a^2=c,c^2=d,e^a=e^2,f^a=f^6,f^b=f^248 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 44);