Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of ΓL(1,𝔽8)⋊Q8 |
| automorphism | Aut(G) = C22×C3×AGL(1,𝔽7) (order 504) · Inn(G) = C2×AGL(1,𝔽7) · Out(G) = C2×C3 |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (C22⋊D7)⋊C9 |
| Hall | C7⋊C9 ⋊ D4 |
| Hall | C22⋊D7 ⋊ 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:
(1,2,7)(3,4,6)(9,11)(12,16,20,14,19,15,17,13,18);(2,3)(4,6)(5,7)(8,9)(10,11)Polycyclic presentation:
2^2,3,2,3,7|c^3=e,b^a=bd,f^a=f^6,f^c=f^457 conjugacy classes — view the full character table.
| maximal subgroup of | #1157⋊C9, D4×D7⋊C9, #1159⋊C9, … |
| maximal normal subgroup of | #1157⋊C9, D4×D7⋊C9, #1159⋊C9, … |
| maximal quotient of | (Dic7⋊C4)⋊C9, ((C2×D7)⋊C4)⋊C9, #1141⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 11);