#72309274

order	16384
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72266130
commutator, Frattini, cosocle	C₂³×H(5,𝔽₂)	C₂⁶
LCS	G ▷ C₂³×H(5,𝔽₂) ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂³×H(5,𝔽₂) ▷ C₂³ ▷ C₂ ▷ C₁

Only the 6133 normal subgroups are shown; non-normal subgroups are omitted.

Full subgroup data →

Presentation

Polycyclic presentation:

2^14|b^a=bg,d^c=dh,e^a=ei,e^c=ej,f^b=fj,f^d=fk,g^f=gl,h^e=hm,i^c=il,i^h=in,j^a=jl,j^d=jm,k^b=km,k^g=kn,l^d=ln,m^a=mn

Conjugacy classes & character table

325 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	UT(6,𝔽₂)
maximal normal subgroup of	UT(6,𝔽₂)

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", "n");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.12);;
gap> SetConjugate(coll, 8, 5, F.8*F.13);;
gap> SetConjugate(coll, 9, 3, F.9*F.12);;
gap> SetConjugate(coll, 9, 8, F.9*F.14);;
gap> SetConjugate(coll, 10, 1, F.10*F.12);;
gap> SetConjugate(coll, 10, 4, F.10*F.13);;
gap> SetConjugate(coll, 11, 2, F.11*F.13);;
gap> SetConjugate(coll, 11, 7, F.11*F.14);;
gap> SetConjugate(coll, 12, 4, F.12*F.14);;
gap> SetConjugate(coll, 13, 1, F.13*F.14);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);