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#72323055

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

Aliases

No aliases for this group.

Subgroups

2998 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^12|c^2=g,e^2=kl,f^2=hj,g^2=i,h^2=jk,j^2=kl,k^2=l,c^a=ck,c^b=ci,d^a=df,d^b=dl,d^c=de,e^a=ek,e^b=el,e^d=ek,f^a=fh,f^c=fkl,f^d=fh,g^a=gl,g^d=gk,g^f=gl,h^a=hj,h^c=hl,h^d=hj,i^d=il,j^a=jk,j^d=jk,k^a=kl,k^d=kl

Conjugacy classes & character table

178 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, 3, F.7);;
gap> SetPower(coll, 5, F.11*F.12);;
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, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.12);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.12);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.11*F.12);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.12);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.12);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.12);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.12);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 4, F.10*F.11);;
gap> SetConjugate(coll, 11, 1, F.11*F.12);;
gap> SetConjugate(coll, 11, 4, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);