Finite Group Databasebeta

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#72307000

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

Aliases

No aliases for this group.

Subgroups

3751 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|a^2=h,b^2=f,e^2=j,f^2=i,i^2=j,b^a=be,c^a=cg,d^b=dghj,e^c=ej,f^a=fj,g^b=gj,h^b=hj

Conjugacy classes & character table

160 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.8);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);