Finite Group Databasebeta

?

← Prev (#72322900) Next (#72322902) →

#72322901

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

Aliases

No aliases for this group.

Subgroups

1684 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^11|c^2=e,d^2=gj,f^2=hi,h^2=ij,i^2=jk,j^2=k,c^a=cf,c^b=cg,d^a=dfk,d^b=dk,d^c=dejk,e^b=ek,f^a=fh,f^c=fh,f^d=fhk,g^c=gk,g^d=gk,h^a=hi,h^c=hi,h^d=hi,i^a=ij,i^c=ij,i^d=ij,j^a=jk,j^c=jk,j^d=jk

Conjugacy classes & character table

140 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.7*F.10);;
gap> SetPower(coll, 6, F.8*F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
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.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.10*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> SetConjugate(coll, 10, 4, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);