Finite Group Databasebeta

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#72309607

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

Aliases

No aliases for this group.

Subgroups

940 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|a^2=e,b^2=f,c^2=g,d^2=e,f^2=h,g^2=i,i^2=j,b^a=bj,c^a=cg,c^b=cfj,d^a=dj,d^b=dh,d^c=dhij,e^c=ej,f^c=fh,g^a=gi,g^d=gj,i^a=ij

Conjugacy classes & character table

82 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.6);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);