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#72322837

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

Aliases

No aliases for this group.

Subgroups

2340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^12|a^2=f,c^2=efh,d^2=g,e^2=hj,f^2=hj,g^2=i,h^2=jk,j^2=kl,k^2=l,b^a=bl,c^a=ce,c^b=ci,d^a=djk,d^c=dj,e^b=el,e^c=eh,e^d=ekl,f^c=fh,f^d=fkl,g^a=gk,g^c=gkl,g^e=gl,g^f=gl,h^c=hj,h^d=hl,i^a=il,i^c=il,j^c=jk,k^c=kl

Conjugacy classes & character table

226 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", "l");;
gap> coll := SingleCollector(F, [2, 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.8);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10*F.11);;
gap> SetPower(coll, 10, F.11*F.12);;
gap> SetPower(coll, 11, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.12);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.10*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.12);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.11*F.12);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.11*F.12);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11*F.12);;
gap> SetConjugate(coll, 7, 5, F.7*F.12);;
gap> SetConjugate(coll, 7, 6, F.7*F.12);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.12);;
gap> SetConjugate(coll, 9, 1, F.9*F.12);;
gap> SetConjugate(coll, 9, 3, F.9*F.12);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> SetConjugate(coll, 11, 3, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);