Finite Group DatabasebetaFinite Group Databasebeta| order | 21560 |
| automorphism | Aut(G) order 36220800 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D10780 |
| Fitting | C4×C5×C11×C49 ⋊ C2 |
| Hall | C5×C11×C49 ⋊ D4 |
| Hall | C11×C49 ⋊ D20 |
| Hall | C5×C49 ⋊ D44 |
| Hall | C5×C11 ⋊ D196 |
| Sylow | C49 ⋊ D220 |
| Sylow | C11 ⋊ D980 |
| Sylow | C5 ⋊ D2156 |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 10780 |
A minimal faithful permutation representation — 2 generators on 69 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)(59,60,61,62,63,64,65,66,67,68,69);(2,4)(6,9)(7,8)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(60,69)(61,68)(62,67)(63,66)(64,65)Polycyclic presentation:
2^3,5,7,11,7|b^2=c,e^7=g,b^a=bc,d^a=d^4,e^a=e^6g^6,f^a=f^10,g^a=g^65393 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D21560, D32340 |
| maximal normal subgroup of | D21560 |
| maximal quotient of | Dic10780, D21560, D32340 |
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, 5, 7, 11, 7]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^6*F.7^6);;
gap> SetConjugate(coll, 6, 1, F.6^10);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);