#72339816

order	8192
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72339650
commutator, Frattini, cosocle	#6374317	C₂⁴
LCS	G ▷ #6374317 ▷ C₂²×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₄² ◁ #72302242 ◁ G
LECS	G ▷ #6374317 ▷ C₂²×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

At least 26775 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^13|c^2=ghil,d^2=hl,e^2=jkl,i^2=l,j^2=m,k^2=m,c^a=cg,c^b=cghi,d^a=djlm,d^b=df,d^c=de,e^a=eh,e^b=ei,e^c=ehjm,e^d=ejk,f^c=fhilm,f^e=fk,g^d=ghlm,g^e=glm,h^c=hl,h^d=hm,h^e=hm,i^b=il,i^c=il,i^d=ik,i^f=il,j^a=jm,j^c=jm,k^b=kl,k^c=km,k^d=km,k^f=km,l^d=lm

Conjugacy classes & character table

137 conjugacy classes — view the full character table.

Relationships

automorphism group of

#28250, #29231

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", "m");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.7*F.8*F.9*F.12);;
gap> SetPower(coll, 4, F.8*F.12);;
gap> SetPower(coll, 5, F.10*F.11*F.12);;
gap> SetPower(coll, 9, F.12);;
gap> SetPower(coll, 10, F.13);;
gap> SetPower(coll, 11, F.13);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.10*F.12*F.13);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10*F.13);;
gap> SetConjugate(coll, 5, 4, F.5*F.10*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9*F.12*F.13);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.8*F.12*F.13);;
gap> SetConjugate(coll, 7, 5, F.7*F.12*F.13);;
gap> SetConjugate(coll, 8, 3, F.8*F.12);;
gap> SetConjugate(coll, 8, 4, F.8*F.13);;
gap> SetConjugate(coll, 8, 5, F.8*F.13);;
gap> SetConjugate(coll, 9, 2, F.9*F.12);;
gap> SetConjugate(coll, 9, 3, F.9*F.12);;
gap> SetConjugate(coll, 9, 4, F.9*F.11);;
gap> SetConjugate(coll, 9, 6, F.9*F.12);;
gap> SetConjugate(coll, 10, 1, F.10*F.13);;
gap> SetConjugate(coll, 10, 3, F.10*F.13);;
gap> SetConjugate(coll, 11, 2, F.11*F.12);;
gap> SetConjugate(coll, 11, 3, F.11*F.13);;
gap> SetConjugate(coll, 11, 4, F.11*F.13);;
gap> SetConjugate(coll, 11, 6, F.11*F.13);;
gap> SetConjugate(coll, 12, 4, F.12*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);