#72323280

order	8192
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72266303
commutator, Frattini, cosocle	C₂×#34980	C₂⁴
LCS	G ▷ C₂×#34980 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#7670203 ◁ G
LECS	G ▷ C₂×#34980 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁

Only the 398 normal subgroups are shown; non-normal subgroups are omitted.

Full subgroup data →

Presentation

Polycyclic presentation:

2^13|a^2=e,c^2=gjl,b^a=be,c^a=cf,c^b=cfg,d^a=dl,d^b=dl,d^c=dh,e^c=ei,e^d=em,f^a=fi,f^b=fj,f^c=fjl,f^d=fl,g^a=gjklm,g^b=gjl,g^c=gl,g^e=gkm,g^f=gl,h^a=hl,h^b=hl,h^e=hm,i^b=ik,i^c=ikm,i^d=im,i^g=im,j^a=jk,j^c=jl,k^c=km,l^a=lm

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323414, #72325772, #72338196
maximal normal subgroup of	#72323414, #72325772, #72338196

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