Finite Group Databasebeta

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#72330236

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

Aliases

No aliases for this group.

Subgroups

At least 7554 conjugacy classes of subgroups; only the 999 normal subgroups are shown.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=h,c^2=fj,d^2=ghij,b^a=bi,c^a=cg,c^b=cf,d^a=dfi,d^b=dij,d^c=dj,e^a=efgi,e^b=eh,e^c=ef,e^d=eghij

Conjugacy classes & character table

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