Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C23 |
| direct product | C22×D7⋊C9 |
| unambiguous semidirect product | D7⋊(C22×C9) |
| unambiguous semidirect product | (C7⋊C9)⋊C23 |
| unambiguous semidirect product | (C2×C7⋊C9)⋊C22 |
| derived, Sylow | C7 ⋊ C23×C9 |
| Hall | C7⋊C9 ⋊ C23 |
| Hall | C22×D7 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | AGL(1,𝔽7) |
| commutator, hypocenter | C7 | C23×C9 |
| Frattini | C3 | C22×AGL(1,𝔽7) |
| Fitting, socle | C22×C3×C7 | C2×C3 |
| prime radical | C22×C3×D7 | C3 |
| cosocle | C3×C7 | C23×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C22×C3 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 252 |
A minimal faithful permutation representation — 3 generators on 20 points:
(6,10,9,7,8,11)(12,16,20,14,19,15,17,13,18);(3,4)(5,6,9,10,8,11,7);(1,2)84 conjugacy classes — view the full character table.
| maximal subgroup of | ((C2×D7)⋊C4)⋊C9, C2×C4×D7⋊C9, C2×C4⋊(D7⋊C9), … |
| maximal normal subgroup of | ((C2×D7)⋊C4)⋊C9, C2×C4×D7⋊C9, C2×C4⋊(D7⋊C9), … |
| maximal quotient of | C2×Dic14⋊C9, C2×C4×D7⋊C9, C2×C4⋊(D7⋊C9), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 56);