Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of ΓL(1,𝔽8)⋊Q8 |
| automorphism | Aut(G) = C3×D4×AGL(1,𝔽7) (order 1008) · Inn(G) = C2×AGL(1,𝔽7) · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊(D7⋊C9) |
| unambiguous semidirect product | D28⋊C9 |
| Hall | C7⋊C9 ⋊ D4 |
| Hall | D28 ⋊ C9 |
| Sylow | C7 ⋊ D4×C9 |
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:
(1,2,3,4)(5,6,11)(7,8,10)(12,16,20,14,19,15,17,13,18);(2,4)(6,7)(8,10)(9,11)Polycyclic presentation:
2^2,3,2,3,7|b^2=d,c^3=e,b^a=bd,f^a=f^6,f^c=f^457 conjugacy classes — view the full character table.
| maximal subgroup of | #1132⋊C9, D56⋊C9, #1141⋊C9, … |
| maximal normal subgroup of | #1132⋊C9, D56⋊C9, #1141⋊C9, … |
| maximal quotient of | #1132⋊C9, D56⋊C9, Dic28⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 9);