Finite Group Databasebeta

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#72340630

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

Aliases

No aliases for this group.

Subgroups

3263 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=g,c^2=j,d^2=ij,e^2=j,f^2=i,h^2=j,d^a=df,d^b=dj,d^c=de,e^a=eh,e^c=ej,f^c=fhi,f^d=fi,f^e=fi,g^d=gi,g^e=gj,h^d=hi,h^e=hj,i^c=ij

Conjugacy classes & character table

88 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, 3, F.10);;
gap> SetPower(coll, 4, F.9*F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);