Finite Group DatabasebetaFinite Group Databasebeta| order | 1200 |
| automorphism | Aut(G) order 96000 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic300 |
| Hall | C3×C25 ⋊ Q16 |
| Sylow | C25 ⋊ Dic12 |
| Sylow | C3 ⋊ Dic100 |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 1200 |
A minimal faithful permutation representation — 2 generators on 44 points:
(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,19)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)Polycyclic presentation:
2^4,3,5^2|a^2=d,b^2=cd,c^2=d,f^5=g,b^a=bc,c^a=cd,e^a=e^2,f^a=f^4g^4,g^a=g^4303 conjugacy classes — view the full character table.
| maximal subgroup of | Dic600, Dic900, Dic1500, … |
| maximal normal subgroup of | Dic600 |
| maximal quotient of | Dic900, Dic1500, Dic2100, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1200, 65);