Finite Group Databasebeta

?

← Prev (#72321785) Next (#72321787) →

#72321786

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

Aliases

No aliases for this group.

Subgroups

429 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|d^2=f,e^2=gh,g^2=hi,h^2=ij,i^2=j,b^a=bf,c^a=ce,d^a=dj,d^b=dj,d^c=di,e^a=eg,e^b=ej,e^c=eg,e^d=ej,f^c=fj,g^a=gh,g^c=gh,h^a=hi,h^c=hi,i^a=ij,i^c=ij

Conjugacy classes & character table

82 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7*F.8);;
gap> SetPower(coll, 7, F.8*F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);