Finite Group Databasebeta

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#72309446

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

Aliases

No aliases for this group.

Subgroups

1495 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|a^2=e,b^2=g,d^2=e,f^2=h,g^2=i,i^2=j,b^a=bg,c^b=cf,d^a=dj,d^b=dij,d^c=dh,e^b=ej,f^b=fh,f^c=fh,g^a=gi,g^d=gj,i^a=ij

Conjugacy classes & character table

85 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.5);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);