Finite Group Databasebeta

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#72264403

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

Aliases

No aliases for this group.

Subgroups

2277 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)(9,10,12,15,13,18,24,30,14,17,26,29,27,34,38,35)(11,19,21,28,23,32,37,40,20,16,25,31,22,33,36,39);(10,15,17,29)(11,20)(12,24)(13,27)(16,31,19,28)(18,35,34,30)(21,36)(25,37)(26,38)(32,39,33,40);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,25)(13,23)(14,20)(15,28)(17,19)(18,33)(21,26)(22,27)(24,36)(29,31)(30,40)(32,34)(35,39)(37,38);(2,6)(5,8)(10,17)(15,29)(16,19)(18,34)(28,31)(30,35)(32,33)(39,40)

Polycyclic presentation:

2^10|a^2=e,b^2=g,c^2=h,e^2=i,f^2=j,i^2=j,b^a=befj,c^a=cg,c^b=cj,d^a=dhj,e^b=ei,e^c=ej,f^b=fj,f^c=fj,g^a=gj,i^b=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.5);;
gap> SetPower(coll, 2, F.7);;
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*F.6*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
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, 1, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);