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#72280648

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

Aliases

No aliases for this group.

Subgroups

18573 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 6 generators on 28 points:

(2,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,22,25,24)(23,27,28,26);(5,7,9,12)(6,10,8,11)(13,15)(14,18)(16,19)(17,20);(1,2,3,4)(7,12)(10,11)(15,20)(18,19)(23,28)(26,27);(5,7,9,12)(6,10,8,11)(13,14,17,16)(15,18,20,19)(21,23,25,28)(22,26,24,27);(5,7)(6,11)(8,10)(9,12)(13,15)(14,19)(16,18)(17,20);(5,8,9,6)(7,11,12,10)(13,16,17,14)(15,19,20,18)(21,24,25,22)(23,26,28,27)

Polycyclic presentation:

2^10|a^2=h,b^2=i,c^2=g,d^2=ghj,f^2=ghj,c^a=cg,c^b=cgj,d^a=dghij,d^c=dhi,e^a=egj,e^b=ei,e^c=egj,e^d=egj,f^a=fgj,f^d=fh,f^e=fgj

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

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