Finite Group Databasebeta

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#72240732

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

Aliases

No aliases for this group.

Subgroups

3621 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=efj,c^2=h,e^2=i,f^2=j,i^2=j,b^a=be,c^a=cgh,c^b=cj,d^b=dh,e^a=ei,e^b=ej,e^c=ej,f^a=fj,f^b=fj,f^c=fj,g^b=gj,i^a=ij

Conjugacy classes & character table

124 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.7);;
gap> SetPower(coll, 2, F.5*F.6*F.10);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);