#72323131

order	4096
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72323126
commutator	C₂²×C₃₂	C₂³×C₄
Frattini, cosocle	C₂×#1397	C₂⁴
prime radical	#72322063	C₂
LCS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ #479917 ◁ #72231861 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

2152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^12|a^2=g,b^2=gkl,c^2=gkl,e^2=kl,f^2=hjkl,g^2=i,h^2=jkl,j^2=kl,k^2=l,b^a=bk,c^a=ck,c^b=cil,d^a=df,d^b=dj,d^c=de,e^a=ek,e^b=el,e^d=ek,f^a=fh,f^b=fkl,f^c=fkl,f^d=fhkl,g^d=gkl,g^f=gl,h^a=hj,h^b=hl,h^c=hl,h^d=hjl,i^d=il,j^a=jk,j^d=jk,k^a=kl,k^d=kl

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323149
maximal normal subgroup of	#72323149

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