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#72322819

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

Aliases

No aliases for this group.

Subgroups

1294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 4 generators on 128 points:

(1,2,7,16,52,56,45,72,58,74,49,78,51,59,113,80,57,84,47,114,13,35,70,33,32,18,48,15,41,17,50,81,11,21,46,82,39,36,37,76,9,19,6,42,40,14,71,73,10,20,54,44,38,83,111,77,8,55,53,79,5,4,31,34)(3,22,26,86,106,94,95,121,110,67,25,100,103,62,120,122,109,68,117,116,69,126,127,91,108,87,118,63,99,23,90,92,30,12,60,64,97,107,102,123,28,88,43,115,98,85,128,124,29,89,104,101,61,93,119,112,27,66,105,75,24,65,96,125);(1,3)(2,12)(4,23)(5,24)(6,43)(7,26)(8,27)(9,28)(10,29)(11,30)(13,69)(14,62)(15,75)(16,64)(17,65)(18,66)(19,67)(20,68)(21,22)(25,49)(31,96)(32,108)(33,112)(34,92)(35,93)(36,94)(37,102)(38,61)(39,97)(40,98)(41,99)(42,100)(44,116)(45,95)(46,60)(47,117)(48,118)(50,90)(51,103)(52,106)(53,105)(54,104)(55,87)(56,107)(57,109)(58,110)(59,85)(63,79)(70,127)(71,128)(72,123)(73,122)(74,88)(76,121)(77,91)(78,115)(80,124)(81,125)(82,86)(83,126)(84,89)(101,114)(111,119)(113,120);(1,4,32,36,10,59,58,83,11,17,8,56,57,14,9,35)(2,13,19,40,84,52,55,41,21,38,74,51,20,39,18,5)(3,23,27,107,29,85,28,93,30,65,108,94,109,62,110,126)(6,44,46,79,48,16,47,78,49,114,7,15,53,82,54,42)(12,61,88,98,89,97,66,99,22,69,67,103,68,106,87,24)(25,101,60,63,105,86,117,115,43,116,26,75,118,64,104,100)(31,81,111,76,71,73,37,77,50,34,70,72,113,80,45,33)(90,92,119,121,120,124,102,91,96,125,127,123,128,122,95,112);(1,5,8,38,10,40,9,39,11,41,32,13,57,51,58,52)(2,14,18,56,20,17,74,83,21,59,55,36,84,4,19,35)(3,24,27,61,29,98,28,97,30,99,108,69,109,103,110,106)(6,45,46,31,48,111,47,71,49,37,7,50,53,70,54,113)(12,62,66,107,68,65,88,126,22,85,87,94,89,23,67,93)(15,76,44,34,78,33,82,73,79,72,114,81,42,77,16,80)(25,102,26,90,105,127,104,120,43,95,60,96,118,119,117,128)(63,123,101,125,100,91,64,124,75,121,116,92,115,112,86,122)

Polycyclic presentation:

2^11|a^2=f,c^2=efgj,d^2=h,e^2=gijk,f^2=ghjk,g^2=ijk,h^2=j,i^2=jk,j^2=k,b^a=bk,c^a=ce,c^b=chi,d^a=djk,d^c=dij,e^a=ejk,e^b=ek,e^c=eg,e^d=ek,f^c=fg,f^d=fk,f^e=fk,g^a=gk,g^c=gi,h^a=hk,h^c=hj,i^c=ij,j^c=jk

Conjugacy classes & character table

188 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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 3, F.5*F.6*F.7*F.10);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.7*F.9*F.10*F.11);;
gap> SetPower(coll, 6, F.7*F.8*F.10*F.11);;
gap> SetPower(coll, 7, F.9*F.10*F.11);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10*F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.10*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);