Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| automorphism | Aut(G) order 138240 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D720 |
| Fitting | C5×C9×C16 ⋊ C2 |
| Hall | C5×C9 ⋊ D16 |
| Sylow | C9 ⋊ D80 |
| Sylow | C5 ⋊ D144 |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 720 |
A minimal faithful permutation representation — 2 generators on 30 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25)(26,27,28,29,30);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,25)(19,24)(20,23)(21,22)(27,30)(28,29)Polycyclic presentation:
2^5,3,5,3|b^2=cd,c^2=de,d^2=e,f^3=h,b^a=bc,c^a=cd,d^a=de,f^a=f^2h^2,g^a=g^4,h^a=h^2363 conjugacy classes — view the full character table.
| maximal subgroup of | D1440, D2160, D3600, … |
| maximal normal subgroup of | D1440 |
| maximal quotient of | D1440, Dic720, D2160, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 160);