Finite Group DatabasebetaFinite Group Databasebeta| order | 6032 |
| automorphism | Aut(G) order 4053504 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D3016 |
| Fitting | C8×C13×C29 ⋊ C2 |
| Hall | C13×C29 ⋊ D8 |
| Sylow | C29 ⋊ D104 |
| Sylow | C13 ⋊ D232 |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 3016 |
A minimal faithful permutation representation — 2 generators on 50 points:
(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50);(2,8)(3,7)(4,6)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(23,50)(24,49)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)Polycyclic presentation:
2^4,13,29|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^12,f^a=f^281511 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D6032, D9048, D15080, … |
| maximal normal subgroup of | D6032 |
| maximal quotient of | D6032, Dic3016, D9048, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 13, 29]);;
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.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^28);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);