Finite Group Databasebeta

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#72234396

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

Aliases

No aliases for this group.

Subgroups

267 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 3 generators on 72 points:

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

Polycyclic presentation:

2^10|a^2=d,b^2=e,c^2=i,d^2=f,e^2=g,f^2=h,g^2=ij,h^2=i,i^2=j,b^a=bi,c^a=cj,d^b=dj,e^a=ej

Conjugacy classes & character table

352 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.4);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.9);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);