Finite Group Databasebeta

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#72307243

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

Aliases

No aliases for this group.

Subgroups

5870 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,5,8)(3,6,4,7)(9,10,13,16,14,18,20,19)(11,15,23,30,21,29,34,27)(12,17,25,31,26,33,24,32)(22,28,37,39,35,40,36,38);(1,3,5,4)(2,6,8,7)(9,11,20,34,14,21,13,23)(10,15,19,27,18,29,16,30)(12,22,24,36,26,35,25,37)(17,28,32,38,33,40,31,39);(2,7)(3,4)(6,8)(10,16,18,19)(11,21)(15,27,29,30)(17,32,33,31)(22,35)(23,34)(28,39,40,38)(36,37);(9,12)(10,17)(11,22)(13,25)(14,26)(15,28)(16,31)(18,33)(19,32)(20,24)(21,35)(23,37)(27,38)(29,40)(30,39)(34,36);(2,8)(6,7)(10,18)(15,29)(16,19)(17,33)(27,30)(28,40)(31,32)(38,39)

Polycyclic presentation:

2^10|a^2=g,b^2=gj,c^2=ghj,f^2=i,g^2=j,h^2=j,c^a=cf,c^b=ci,d^c=dghj,e^a=ei,f^a=fi,f^c=fij,f^d=fj,h^a=hj

Conjugacy classes & character table

148 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.7);;
gap> SetPower(coll, 2, F.7*F.10);;
gap> SetPower(coll, 3, F.7*F.8*F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);