Finite Group Databasebeta

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#72328581

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

Aliases

No aliases for this group.

Subgroups

At least 9294 conjugacy classes of subgroups; only the 3017 normal subgroups are shown.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^10|c^2=h,e^2=g,f^2=i,b^a=bj,c^a=ci,c^b=cg,d^c=dh,e^a=ei,e^b=eh,e^c=egh,e^d=eh,f^a=fi,f^b=fh,f^d=fi,f^e=fghij

Conjugacy classes & character table

118 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, 3, F.8);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);