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#72318567

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

Aliases

No aliases for this group.

Subgroups

3334 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|b^2=i,c^2=hj,b^a=bh,c^b=cf,d^a=dfg,d^b=dfghij,d^c=di,e^a=egh,e^d=ei,f^c=fj,f^d=fj,g^b=gj,g^d=gj,h^b=hj,i^a=ij

Conjugacy classes & character table

82 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.9);;
gap> SetPower(coll, 3, F.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, 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);