Finite Group DatabasebetaFinite Group Databasebeta| order | 4680 |
| automorphism | Aut(G) order 1347840 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D2340 |
| Fitting | C4×C5×C9×C13 ⋊ C2 |
| Hall | C5×C9×C13 ⋊ D4 |
| Hall | C9×C13 ⋊ D20 |
| Hall | C5×C13 ⋊ D36 |
| Hall | C5×C9 ⋊ D52 |
| Sylow | C13 ⋊ D180 |
| Sylow | C9 ⋊ D260 |
| Sylow | C5 ⋊ D468 |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 2340 |
A minimal faithful permutation representation — 2 generators on 31 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);(2,4)(6,13)(7,12)(8,11)(9,10)(15,18)(16,17)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)Polycyclic presentation:
2^3,3,5,13,3|b^2=c,d^3=g,b^a=bc,d^a=d^2g^2,e^a=e^4,f^a=f^12,g^a=g^21173 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D4680, D7020, D11700, … |
| maximal normal subgroup of | D4680 |
| maximal quotient of | D4680, D7020, Dic2340, … |
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, 3, 5, 13, 3]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.7^2);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);