Finite Group DatabasebetaFinite Group Databasebeta| order | 9520 |
| automorphism | Aut(G) order 7311360 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D4760 |
| Fitting | C5×C7×C8×C17 ⋊ C2 |
| Hall | C5×C7×C17 ⋊ D8 |
| Hall | C7×C17 ⋊ D40 |
| Hall | C5×C17 ⋊ D56 |
| Hall | C5×C7 ⋊ D136 |
| Sylow | C17 ⋊ D280 |
| Sylow | C7 ⋊ D680 |
| Sylow | C5 ⋊ D952 |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 4760 |
A minimal faithful permutation representation — 2 generators on 37 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);(2,8)(3,7)(4,6)(10,13)(11,12)(15,20)(16,19)(17,18)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)Polycyclic presentation:
2^4,5,7,17|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^4,f^a=f^6,g^a=g^162383 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D9520, D14280, D23800 |
| maximal normal subgroup of | D9520 |
| maximal quotient of | D9520, Dic4760, D14280, … |
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, 5, 7, 17]);;
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.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 7, 1, F.7^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);