Finite Group Databasebeta

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#72280752

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

Aliases

No aliases for this group.

Subgroups

4873 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=gij,d^2=f,e^2=j,f^2=j,h^2=i,b^a=bg,d^a=de,d^b=dh,d^c=dj,e^a=ei,e^b=ej,f^a=fj,f^b=fi,g^d=gij,h^a=hi,h^b=hj

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

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