Finite Group DatabasebetaFinite Group Databasebeta| order | 5184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2 |
No aliases for this group.
1045 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 1045 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 44 points:
(1,2,6,11)(3,7,12,8)(4,23,18,24,15,10,16,25,5,9,17,26)(13,31,28,21,29,20,27,32,14,19,30,22)(33,34,35)(36,39,44,38,42,40,41,37,43);(1,17,6,15)(2,32,11,31)(3,29,12,30)(4,5,16,18)(7,25,8,23)(9,20,24,22)(10,21,26,19)(13,28,27,14)(34,35)(37,39,42)(40,44,43)Polycyclic presentation:
2^2,3^2,2,3,2^2,3,2|b^2=j,d^3=f,e^2=j,g^2=j,h^2=j,b^a=be,d^c=df,e^a=ej,e^b=ej,g^d=h,h^d=ghj,h^g=hj,i^a=i^2213 conjugacy classes — view the full character table.
| maximal subgroup of | #72253014 |
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", "j");;
gap> coll := SingleCollector(F, [2, 2, 3, 3, 2, 3, 2, 2, 3, 2]);;
gap> SetPower(coll, 2, F.10);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 8, 4, F.7*F.8*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);