Finite Group DatabasebetaFinite Group Databasebeta| order | 7568 |
| automorphism | Aut(G) order 6357120 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D3784 |
| Fitting | C8×C11×C43 ⋊ C2 |
| Hall | C11×C43 ⋊ D8 |
| Sylow | C43 ⋊ D88 |
| Sylow | C11 ⋊ D344 |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 62 |
| transitive degree | μt(G) = 3784 |
A minimal faithful permutation representation — 2 generators on 62 points:
(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62);(2,8)(3,7)(4,6)(10,19)(11,18)(12,17)(13,16)(14,15)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)Polycyclic presentation:
2^4,11,43|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^10,f^a=f^421895 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D7568, D11352, D18920, … |
| maximal normal subgroup of | D7568 |
| maximal quotient of | D7568, Dic3784, D11352, … |
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, 11, 43]);;
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.5^10);;
gap> SetConjugate(coll, 6, 1, F.6^42);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);