Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic180 |
| Hall | C5×C9 ⋊ Q16 |
| Sylow | C9 ⋊ Dic20 |
| Sylow | C5 ⋊ Dic36 |
54 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)(9,16,15,14,13,12,11,10)(17,18,19,20,21,22,23,24,25)(26,27,28,29,30);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,25)(19,24)(20,23)(21,22)(27,30)(28,29)Polycyclic presentation:
2^4,3,5,3|a^2=d,b^2=cd,c^2=d,e^3=g,b^a=bc,c^a=cd,e^a=e^2g^2,f^a=f^4,g^a=g^2183 conjugacy classes — view the full character table.
| maximal subgroup of | #13083695, #13083697, #13083703, … |
| maximal normal subgroup of | #13083695, #13083697, #13083703, … |
| maximal quotient of | #13083861, #13083862, C2×Dic180, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 65);