Finite Group Databasebeta

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#72307960

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

Aliases

No aliases for this group.

Subgroups

3862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=g,c^2=fij,e^2=i,f^2=ij,i^2=j,b^a=bgh,c^a=ce,c^b=cefi,d^c=dhj,e^a=ei,f^a=fij,f^b=fj,g^c=gj,i^a=ij

Conjugacy classes & character table

100 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, 3, F.6*F.9*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);