Finite Group Databasebeta

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#72307302

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

Aliases

No aliases for this group.

Subgroups

3807 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|a^2=h,c^2=hi,e^2=ij,h^2=j,i^2=j,b^a=bf,c^a=ci,d^a=dg,d^b=dghij,e^a=egi,e^c=ej,f^d=fj,i^a=ij

Conjugacy classes & character table

124 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.8);;
gap> SetPower(coll, 3, F.8*F.9);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);