Finite Group Databasebeta

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#72234377

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

Aliases

No aliases for this group.

Subgroups

400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 80 points:

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

Polycyclic presentation:

2^11|a^2=c,b^2=d,c^2=f,d^2=g,e^2=hk,f^2=i,g^2=k,h^2=k,i^2=j,j^2=k,b^a=be,c^b=ch,d^a=dhk,d^c=dk,f^b=fk,g^a=gk

Conjugacy classes & character table

368 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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.11);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetPower(coll, 9, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);