Finite Group Databasebeta

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#72250002

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

Aliases

No aliases for this group.

Subgroups

2143 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^10|a^2=f,b^2=fh,c^2=fh,d^2=g,e^2=j,g^2=i,i^2=j,c^a=cfhj,d^a=de,d^b=dg,d^c=di,e^b=ej,e^c=ej,f^d=fj,g^a=gj,g^b=gi,g^c=gj,i^b=ij

Conjugacy classes & character table

94 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.6);;
gap> SetPower(coll, 2, F.6*F.8);;
gap> SetPower(coll, 3, F.6*F.8);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
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);