Finite Group Databasebeta

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#72300950

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

Aliases

No aliases for this group.

Subgroups

At least 8936 conjugacy classes of subgroups; only the 2981 normal subgroups are shown.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|c^2=h,e^2=h,b^a=bi,c^b=ch,d^b=dg,d^c=dj,e^b=eh,e^d=ehij,f^a=fhij,f^b=fg,f^c=fij,f^e=fg

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

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