Finite Group Databasebeta

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#72231877

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

Aliases

No aliases for this group.

Subgroups

257 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 3 generators on 80 points:

(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(17,18,20,28,22,30,44,62,24,32,45,64,23,54,21,48,26,34,47,66,53,68,49,29,25,33,46,65,52,31,50,63)(19,35,37,70,39,55,73,80,41,67,76,79,74,60,77,61,36,69,72,57,51,27,38,56,42,58,43,78,40,59,75,71);(19,36)(27,55)(35,69)(37,72)(38,73)(39,51)(40,74)(41,42)(43,76)(56,80)(57,70)(58,67)(59,60)(61,71)(75,77)(78,79);(17,19,21,38,23,40,46,72,25,42,44,77,22,51,47,76,26,36,50,73,52,74,45,37,24,41,49,75,53,39,20,43)(18,27,29,57,31,35,66,71,33,60,48,79,68,58,64,56,34,55,62,70,54,69,28,61,32,59,63,78,30,67,65,80)

Polycyclic presentation:

2^10|a^2=d,c^2=e,d^2=f,e^2=g,f^2=h,g^2=i,h^2=j,i^2=j,c^a=cg,c^b=cj,d^c=di,e^a=ei,e^d=ej,f^c=fj,g^a=gj

Conjugacy classes & character table

184 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, 3, 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, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);