Finite Group Databasebeta

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#72306973

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

Aliases

No aliases for this group.

Subgroups

3942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|b^2=h,c^2=fi,e^2=j,f^2=ij,i^2=j,b^a=bg,c^a=cf,c^b=ce,d^c=dgh,f^a=fi,f^b=fj,g^c=gj,h^c=hj,i^a=ij

Conjugacy classes & character table

100 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.8);;
gap> SetPower(coll, 3, F.6*F.9);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);