Finite Group Databasebeta

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#72319837

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

Aliases

No aliases for this group.

Subgroups

3219 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|a^2=e,c^2=e,h^2=i,i^2=j,b^a=bh,c^b=cij,d^a=dg,d^b=df,d^c=dj,e^b=ej,h^a=hi,h^b=hij,h^c=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

112 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.5);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);