Finite Group Databasebeta

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#72336554

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

Aliases

No aliases for this group.

Subgroups

At least 10232 conjugacy classes of subgroups; only the 3333 normal subgroups are shown.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|a^2=g,e^2=ghj,f^2=hij,b^a=bh,c^a=cgj,c^b=ci,d^a=dghj,e^a=ehj,e^c=egj,e^d=eghj,f^d=fgi,f^e=fghj

Conjugacy classes & character table

124 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.7);;
gap> SetPower(coll, 5, F.7*F.8*F.10);;
gap> SetPower(coll, 6, F.8*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);