Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | Aut(G) = C22×C3×AGL(1,𝔽7) (order 504) · Inn(G) = AGL(1,𝔽7) · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C2 |
| direct product | C4×D7⋊C9 |
| unambiguous semidirect product | D7⋊(C4×C9) |
| unambiguous semidirect product | (C4×D7)⋊C9 |
| derived, Sylow | C7 ⋊ C2×C4×C9 |
| Hall | C7⋊C9 ⋊ C2×C4 |
| Hall | C4×D7 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | AGL(1,𝔽7) |
| commutator, hypocenter | C7 | C2×C4×C9 |
| Frattini | C2×C3 | C2×AGL(1,𝔽7) |
| Fitting | C3×C4×C7 | C2×C3 |
| socle, cosocle | C2×C3×C7 | C22×C3 |
| prime radical | C2×C3×D7 | C2×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C3×C4 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 252 |
A minimal faithful permutation representation — 2 generators on 20 points:
(6,10,9,7,8,11)(12,19,18,14,13,20,17,16,15);(1,2,3,4)(5,6,9,10,8,11,7)84 conjugacy classes — view the full character table.
| maximal subgroup of | C8×D7⋊C9, #1131⋊C9, C2×C4×D7⋊C9, … |
| maximal normal subgroup of | C8×D7⋊C9, #1131⋊C9, C2×C4×D7⋊C9, … |
| maximal quotient of | C8×D7⋊C9, #1131⋊C9, C4×Dic7⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 8);