Finite Group Databasebeta

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#72240736

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

Aliases

No aliases for this group.

Subgroups

4073 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|b^2=ef,c^2=h,e^2=ij,i^2=j,b^a=be,c^a=cg,d^b=dh,e^a=ei,e^b=ej,e^c=ej,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, 2, F.5*F.6);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 5, F.9*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);;
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, 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);