Finite Group DatabasebetaFinite Group Databasebeta| order | 2640 |
| automorphism | Aut(G) order 422400 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic660 |
| Hall | C3×C5×C11 ⋊ Q16 |
| Hall | C5×C11 ⋊ Dic12 |
| Hall | C3×C11 ⋊ Dic20 |
| Hall | C3×C5 ⋊ Dic44 |
| Sylow | C11 ⋊ Dic60 |
| Sylow | C5 ⋊ Dic132 |
| Sylow | C3 ⋊ Dic220 |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 2640 |
A minimal faithful permutation representation — 2 generators on 35 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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,19)(21,24)(22,23)(26,35)(27,34)(28,33)(29,32)(30,31)Polycyclic presentation:
2^4,3,5,11|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^4,g^a=g^10663 conjugacy classes — view the full character table.
| maximal subgroup of | Dic1320, Dic1980, Dic3300, … |
| maximal normal subgroup of | Dic1320 |
| maximal quotient of | Dic1980, Dic3300, Dic4620, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 5, 11]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);