Finite Group DatabasebetaFinite Group Databasebeta| order | 10080 |
| automorphism | Aut(G) order 5806080 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D5040 |
| Fitting | C5×C7×C9×C16 ⋊ C2 |
| Hall | C5×C7×C9 ⋊ D16 |
| Hall | C7×C9 ⋊ D80 |
| Hall | C5×C9 ⋊ D112 |
| Hall | C5×C7 ⋊ D144 |
| Sylow | C9 ⋊ D560 |
| Sylow | C7 ⋊ D720 |
| Sylow | C5 ⋊ D1008 |
168 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 5040 |
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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,25)(19,24)(20,23)(21,22)(27,30)(28,29)(32,37)(33,36)(34,35)Polycyclic presentation:
2^5,3,5,7,3|b^2=cd,c^2=de,d^2=e,f^3=i,b^a=bc,c^a=cd,d^a=de,f^a=f^2i^2,g^a=g^4,h^a=h^6,i^a=i^22523 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D10080, D15120, D25200 |
| maximal normal subgroup of | D10080 |
| maximal quotient of | D10080, Dic5040, D15120, … |
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", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 5, 7, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.9^2);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);