Finite Group Databasebeta

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#72320937

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

Aliases

No aliases for this group.

Subgroups

3004 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,e^2=h,f^2=h,g^2=i,i^2=j,b^a=bgij,c^b=cj,d^a=dj,e^a=eg,e^b=ej,f^a=fj,g^b=gi,g^e=gij,i^b=ij,i^e=ij

Conjugacy classes & character table

184 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.6);;
gap> SetPower(coll, 5, F.8);;
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.7*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 5, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);