Finite Group Databasebeta

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#72227012

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

Aliases

No aliases for this group.

Subgroups

913 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,7,5,8,13,14,6,9,3,10,11,15,12,16)(17,18)(19,28,30,41)(20,33)(21,35)(22,25)(23,27)(24,32,38,29)(26,37)(31,45,44,42)(34,40)(36,47)(39,43,48,46);(17,19,21,31,22,32,36,46,23,30,37,44,20,29,34,43)(18,24,26,39,25,28,40,42,27,38,35,48,33,41,47,45);(17,20,23,22)(18,25,27,33)(19,29,30,32)(21,34,37,36)(24,28,38,41)(26,40,35,47)(31,43,44,46)(39,42,48,45);(1,3,5,12,6,4,11,13)(2,7,8,14,9,10,15,16)

Polycyclic presentation:

2^10|a^2=e,b^2=f,c^2=j,d^2=g,e^2=g,f^2=h,g^2=i,h^2=j,b^a=bh,c^a=cj,d^a=di,e^b=ej,f^a=fj

Conjugacy classes & character table

256 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, 2, F.6);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);