Finite Group Databasebeta

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#72265039

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

Aliases

No aliases for this group.

Subgroups

3245 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^10|b^2=f,c^2=f,e^2=g,g^2=i,i^2=j,b^a=bh,d^b=dj,e^a=eh,e^b=eg,e^c=ehi,e^d=ej,f^e=fj,g^b=gi,g^c=gj,i^b=ij

Conjugacy classes & character table

112 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, 2, F.6);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);