Finite Group DatabasebetaFinite Group Databasebeta| order | 2640 |
| automorphism | Aut(G) order 422400 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D1320 |
| Fitting | C3×C5×C8×C11 ⋊ C2 |
| Hall | C3×C5×C11 ⋊ D8 |
| Hall | C5×C11 ⋊ D24 |
| Hall | C3×C11 ⋊ D40 |
| Hall | C3×C5 ⋊ D88 |
| Sylow | C11 ⋊ D120 |
| Sylow | C5 ⋊ D264 |
| Sylow | C3 ⋊ D440 |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 1320 |
A minimal faithful permutation representation — 2 generators on 27 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);(2,8)(3,7)(4,6)(10,11)(13,16)(14,15)(18,27)(19,26)(20,25)(21,24)(22,23)Polycyclic presentation:
2^4,3,5,11|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^4,g^a=g^10663 conjugacy classes — view the full character table.
| maximal subgroup of | D2640, D3960, D6600, … |
| maximal normal subgroup of | D2640 |
| maximal quotient of | D2640, D3960, Dic1320, … |
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, 3, 5, 11]);;
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.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);