Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of #6706 |
| automorphism | Aut(G) = D3×AGL(1,𝔽5)×AGL(1,ℤ/8) (order 3840) · Inn(G) = D60 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic60 |
| Hall | C3×C5 ⋊ Q16 |
| Sylow | C5 ⋊ Dic12 |
| Sylow | C3 ⋊ Dic20 |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 2 generators on 24 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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,19)(21,24)(22,23)Polycyclic presentation:
2^4,3,5|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^463 conjugacy classes — view the full character table.
| maximal subgroup of | #90942, #90944, #90947, … |
| maximal normal subgroup of | #90942, #90944, #90947, … |
| maximal quotient of | #91100, #91102, C2×Dic60, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 69);