Finite Group Databasebeta

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#72314665

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

Aliases

No aliases for this group.

Subgroups

3191 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|a^2=f,b^2=e,b^a=bfg,c^a=cgi,c^b=cefhi,d^a=dj,d^c=dj,e^a=eh,e^c=ehi,f^b=fhi,f^c=fhij,f^e=fj,g^a=ghi,g^b=gi,h^a=hj,h^b=hj,i^a=ij

Conjugacy classes & character table

58 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.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);