Finite Group Databasebeta

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#72264608

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

Aliases

No aliases for this group.

Subgroups

2362 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=gh,c^2=egij,e^2=i,f^2=j,i^2=j,b^a=bgh,c^a=cf,c^b=ce,d^c=dhj,e^a=ej,e^b=ei,e^c=ej,g^c=gj,i^b=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, 1, F.7);;
gap> SetPower(coll, 2, F.7*F.8);;
gap> SetPower(coll, 3, F.5*F.7*F.9*F.10);;
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.7*F.8);;
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.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);