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) = C1 |
| unambiguous semidirect product | Dic14⋊C9 |
| unambiguous semidirect product | (C7⋊C9)⋊Q8 |
| Hall | C7⋊C9 ⋊ Q8 |
| Hall | Dic14 ⋊ C9 |
| Sylow | C7 ⋊ Q8×C9 |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 504 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,7)(3,4,6)(8,10,11,15)(9,12,13,14)(16,20,24,18,23,19,21,17,22);(2,3)(4,6)(5,7)(8,9,11,13)(10,14,15,12)Polycyclic presentation:
2^2,3,2,3,7|a^2=d,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, Dic28⋊C9, #1142⋊C9, … |
| maximal normal subgroup of | #1132⋊C9, Dic28⋊C9, #1142⋊C9, … |
| maximal quotient of | (Dic7⋊C4)⋊C9, C4⋊(Dic7⋊C9), C2×Dic14⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 7);