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#72319114

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

Aliases

No aliases for this group.

Subgroups

2895 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|b^2=f,c^2=i,d^2=i,b^a=bf,c^a=cfg,d^a=dfh,d^b=dfgh,d^c=dh,e^a=eh,e^b=ei,f^d=fj,g^c=gj,g^d=gj,h^b=hj,i^a=ij

Conjugacy classes & character table

76 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, 2, F.6);;
gap> SetPower(coll, 3, F.9);;
gap> SetPower(coll, 4, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*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);