Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | Aut(G) = C3×D4×AGL(1,𝔽7) (order 1008) · Inn(G) = AGL(1,𝔽7) · Out(G) = C3×D4 |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C2 |
| direct product | C2×Dic7⋊C9 |
| unambiguous semidirect product | (C2×Dic7)⋊C9 |
| unambiguous semidirect product | (C2×C7⋊C9)⋊C4 |
| derived, Sylow | C7 ⋊ C2×C4×C9 |
| Hall | C7⋊C9 ⋊ C2×C4 |
| Hall | C2×Dic7 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | AGL(1,𝔽7) |
| commutator, hypocenter | C7 | C2×C4×C9 |
| Frattini | C2×C3 | C2×AGL(1,𝔽7) |
| Fitting, socle, prime radical | C22×C3×C7 | C2×C3 |
| cosocle | C2×C3×C7 | C22×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C22×C3 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 504 |
A minimal faithful permutation representation — 2 generators on 22 points:
(3,4,5,6)(8,12,11,9,10,13)(14,21,20,16,15,22,19,18,17);(1,2)(7,8,11,12,10,13,9)84 conjugacy classes — view the full character table.
| maximal subgroup of | C4×Dic7⋊C9, (Dic7⋊C4)⋊C9, C4⋊(Dic7⋊C9), … |
| maximal normal subgroup of | C4×Dic7⋊C9, (Dic7⋊C4)⋊C9, C4⋊(Dic7⋊C9), … |
| maximal quotient of | C2×(C7⋊C8)⋊C9, #1136⋊C9, C4×Dic7⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 10);