#72266317

order	4096
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#72214549
commutator, Frattini, cosocle	C₂³×#120	C₂⁴
LCS	G ▷ C₂³×#120 ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×D₄² ◁ G
LECS	G ▷ C₂³×#120 ▷ C₂⁴ ▷ C₂² ▷ C₁

At least 13349 conjugacy classes of subgroups; only the 437 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^12|b^a=be,c^a=cf,c^b=cg,d^a=dh,d^c=di,e^c=ei,f^b=fij,f^e=fk,g^a=gjkl,g^d=gl,g^e=gl,h^c=hk,i^a=ik,i^b=il,j^b=jl

Conjugacy classes & character table

151 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266323, #72266344, #72266691
maximal normal subgroup of	#72266323, #72266344, #72266691
maximal quotient of	#72323840

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