#72323185

order	4096
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72323184
commutator	C₂×C₃₂	C₂×C₄×C₈
Frattini, cosocle	C₂×#7330	C₂³
prime radical	C₂×#41	#1333
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ #479899 ◁ #72231854 ◁ G
LECS	G ▷ C₂×#7330 ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1084 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 128 points:

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

Polycyclic presentation:

2^12|a^2=d,c^2=efhjk,d^2=g,e^2=h,f^2=h,g^2=i,h^2=jl,i^2=k,j^2=k,k^2=l,b^a=bl,c^a=ce,c^b=cijkl,d^c=dh,e^a=ejk,e^b=el,e^c=ehjk,e^d=ek,f^a=fjk,f^c=fhjk,f^d=fk,g^c=gj,g^e=gl,g^f=gl,h^a=hkl,h^c=hjk,h^d=hl,i^c=ik,j^a=jl,j^c=jkl,k^c=kl

Conjugacy classes & character table

226 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323191
maximal normal subgroup of	#72323191

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