Finite Group Databasebeta

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#72271750

• supersolvable
• metabelian
• solvable
• monomial
• stem

Aliases

No aliases for this group.

Splittings

Subgroups

4730 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,3)(4,5)(6,8,11,10)(7,9)(12,13)(14,20,21,26)(15,19)(16,18)(17,22,25,27)(23,24);(4,6)(5,8,9,10)(7,11)(12,14,15,22,16,21,24,27)(13,17,19,20,18,25,23,26);(2,3)(4,5)(6,10)(7,9)(8,11)(12,13)(14,17,21,25)(15,23)(16,18)(19,24)(20,22,26,27)

Polycyclic presentation:

2^9,3|a^2=g,b^2=f,c^2=dg,f^2=i,b^a=be,c^a=cd,c^b=cdfg,d^b=dh,e^a=ei,e^b=ehi,e^c=ei,f^a=fhi,f^c=fh,g^b=gi,j^a=j^2

Conjugacy classes & character table

108 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, 3]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.4*F.7);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);