#72262293

order	8192
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72237720
commutator, Frattini, cosocle	C₂×#2308	C₂⁵
LCS	G ▷ C₂×#2308 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#7670203 ◁ G
LECS	G ▷ C₂×#2308 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

Only the 1290 normal subgroups are shown; non-normal subgroups are omitted.

Full subgroup data →

Presentation

Polycyclic presentation:

2^13|b^2=k,h^2=k,b^a=bk,c^a=ci,c^b=ci,d^b=dg,d^c=dh,e^a=ej,e^b=ej,e^d=ef,f^a=fl,f^c=fj,g^c=gk,g^e=gl,h^a=hk,h^c=hk,h^d=hk,h^e=hjlm,h^f=hlm,i^d=ik,i^f=im,j^b=jm,j^d=jl,j^h=jm,k^e=km,l^c=lm

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323414, #72325772, #72338102
maximal normal subgroup of	#72323414, #72325772, #72338102
automorphism group of	#1781

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