Finite Group Databasebeta

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#72332265

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

Aliases

No aliases for this group.

Subgroups

At least 9336 conjugacy classes of subgroups; only the 3389 normal subgroups are shown.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|b^2=h,c^2=i,d^2=hij,e^2=h,b^a=bj,c^b=ci,d^b=dij,d^c=dhj,e^c=ehi,e^d=eij,f^a=fi,f^c=fi,f^d=fg

Conjugacy classes & character table

148 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, 2, F.8);;
gap> SetPower(coll, 3, F.9);;
gap> SetPower(coll, 4, F.8*F.9*F.10);;
gap> SetPower(coll, 5, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);