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#72322654

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

Aliases

No aliases for this group.

Subgroups

1589 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^11|c^2=e,d^2=h,e^2=h,f^2=gi,g^2=ij,h^2=k,i^2=jk,j^2=k,c^a=cf,c^b=chjk,d^a=dj,e^a=ej,e^b=ek,f^a=fg,f^c=fgj,f^d=fk,f^e=fk,g^a=gi,g^c=gik,h^a=hk,i^a=ij,i^c=ij,j^a=jk,j^c=jk

Conjugacy classes & character table

152 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, 3, F.5);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.7*F.9);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.11);;
gap> SetPower(coll, 9, F.10*F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.10*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.10);;
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.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);