#72323173

order	4096
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72323154
commutator	C₂²×C₃₂	C₂²×C₈
Frattini, cosocle	C₂×#7330	C₂³
prime radical	#512951	C₈
LCS	G ▷ C₂²×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

998 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:

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

Polycyclic presentation:

2^12|a^2=d,b^2=i,d^2=g,e^2=h,f^2=h,g^2=i,h^2=j,j^2=k,k^2=l,b^a=bi,c^a=ce,c^b=cefhjkl,d^c=djkl,e^a=eh,e^b=el,e^c=ehjkl,e^d=ek,f^a=fh,f^c=fhjkl,f^d=fk,g^c=gkl,g^e=gl,g^f=gl,h^a=hj,h^c=hjkl,h^d=hl,i^c=il,j^a=jk,j^c=jkl,k^a=kl,k^c=kl

Conjugacy classes & character table

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