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#72231876

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

Aliases

No aliases for this group.

Subgroups

885 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 4 generators on 72 points:

(2,5)(6,8)(9,10,13,21,15,23,37,54,17,25,35,55,14,41,38,53,18,26,40,51,42,56,36,20,16,24,39,52,43,22,12,34)(11,19,29,46,31,48,64,67,33,50,62,72,30,66,57,71,27,44,58,68,60,69,63,45,32,49,65,70,59,47,28,61);(9,11)(10,19)(12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,27)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,44)(34,61)(35,62)(36,63)(37,64)(38,57)(39,65)(40,58)(41,66)(42,60)(43,59)(51,68)(52,70)(53,71)(54,67)(55,72)(56,69);(11,27)(19,44)(28,57)(29,58)(30,59)(31,60)(32,33)(45,67)(46,68)(47,66)(48,69)(49,50)(61,71)(62,65)(63,64)(70,72);(1,2,3,6,4,5,7,8)(9,12,14,35,16,36,15,13,18,38,43,39,17,37,42,40)(10,20,22,51,24,53,56,55,26,54,41,21,25,34,23,52)(11,28,30,62,32,63,31,29,27,57,59,65,33,64,60,58)(19,45,47,68,49,71,69,72,44,67,66,46,50,61,48,70)

Polycyclic presentation:

2^10|a^2=e,d^2=f,e^2=g,f^2=h,g^2=i,h^2=j,i^2=j,c^b=cj,d^a=dhj,e^d=ej,f^a=fj

Conjugacy classes & character table

328 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, 1, F.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);