Finite Group Databasebeta

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#72286907

• p-group
• nilpotent
• supersolvable
• metabelian
• solvable
• monomial
• stem

Aliases

No aliases for this group.

Subgroups

549 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4);(6,12,14,32,16,34,18,42)(7,19,21,51)(8,24)(9,28)(10,29)(13,36,38,58)(15,37,43,60,45,67,47,68)(17,49,44,66,48,64,50,35)(20,53)(22,40,56,41)(23,39,57,52)(25,30)(26,55)(27,54)(31,59,46,33,61,63,62,65);(2,4)(5,6,13,37,27,45,7,12,25,44,51,65,20,31,36,64,10,18,40,60,53,43,23,32,9,17,39,59,54,62,22,35,11,16,38,67,8,15,21,34,28,50,19,33,55,61,58,49,29,14,41,68,26,47,57,42,30,48,52,63,24,46,56,66)

Polycyclic presentation:

2^11|b^2=f,c^2=dgi,d^2=gij,f^2=h,g^2=ijk,i^2=jk,j^2=k,c^a=ce,c^b=cd,d^b=dg,f^c=fj,f^d=fk,g^b=gi,h^c=hk,i^b=ij,j^b=jk

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

GAP

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.4*F.7*F.9);;
gap> SetPower(coll, 4, F.7*F.9*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10*F.11);;
gap> SetPower(coll, 9, F.10*F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);