#72334693

order	16384
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#72265826
commutator, Frattini, cosocle	C₂⁴×#120	C₂⁵
LCS	G ▷ C₂⁴×#120 ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂⁴×#120 ◁ G
LECS	G ▷ C₂⁴×#120 ▷ C₂⁵ ▷ C₂² ▷ C₁

Only the 1541 normal subgroups are shown; non-normal subgroups are omitted.

Full subgroup data →

Presentation

Polycyclic presentation:

2^14|b^a=bf,c^b=cg,d^c=dh,e^c=ei,f^c=fj,g^a=gj,g^d=gk,g^e=gl,h^b=hk,h^f=hm,i^b=il,i^f=in,j^d=jm,j^e=jn,k^a=km,l^a=ln

Conjugacy classes & character table

280 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.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.12);;
gap> SetConjugate(coll, 8, 2, F.8*F.11);;
gap> SetConjugate(coll, 8, 6, F.8*F.13);;
gap> SetConjugate(coll, 9, 2, F.9*F.12);;
gap> SetConjugate(coll, 9, 6, F.9*F.14);;
gap> SetConjugate(coll, 10, 4, F.10*F.13);;
gap> SetConjugate(coll, 10, 5, F.10*F.14);;
gap> SetConjugate(coll, 11, 1, F.11*F.13);;
gap> SetConjugate(coll, 12, 1, F.12*F.14);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);