Finite Group Databasebeta

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#72309713

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

Aliases

No aliases for this group.

Subgroups

2617 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|a^2=f,b^2=h,c^2=f,d^2=gj,e^2=h,g^2=i,i^2=j,b^a=bj,c^a=cj,c^b=ch,d^a=dg,d^b=dh,d^c=di,e^b=eh,f^d=fj,g^a=gi,g^c=gj,i^a=ij

Conjugacy classes & character table

115 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, 2, F.8);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7*F.10);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
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);