Finite Group Databasebeta

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#72308798

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

Aliases

No aliases for this group.

Subgroups

1405 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 4 generators on 40 points:

(2,6)(3,4)(7,8)(11,19,21,32)(12,23)(13,26)(15,27,28,40)(17,30)(18,29)(20,31,34,35)(22,37,36,33)(24,38)(25,39);(2,7)(3,4)(6,8)(15,28)(20,34)(27,40)(31,35);(9,10)(11,20,21,34)(12,24)(13,18)(14,16)(15,22,28,36)(17,25)(19,31,32,35)(23,38)(26,29)(27,37,40,33)(30,39);(1,2,4,6,5,8,3,7)(9,11,12,19,13,22,23,33,14,21,25,32,26,36,39,37)(10,15,17,27,18,20,30,35,16,28,24,40,29,34,38,31)

Polycyclic presentation:

2^10|a^2=f,c^2=f,d^2=g,e^2=h,g^2=i,i^2=j,c^b=cf,d^a=dg,d^b=de,d^c=dhi,e^a=eh,e^b=eh,e^c=ej,f^d=fj,g^a=gi,g^b=gh,g^c=gj,i^a=ij

Conjugacy classes & character table

70 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);