#72306287

order	4096
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁵×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁷
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×#525 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

Only the 44103 normal subgroups are shown; non-normal subgroups are omitted.

Full subgroup data →

Presentation

Polycyclic presentation:

2^12|b^2=h,c^2=hi,d^2=h,g^2=ij,k^2=l,c^a=chi,c^b=ck,d^c=dl,g^a=ghij,g^b=gij,g^c=gjk,g^d=gh,g^e=gij,g^f=gh,k^b=kl,k^c=kl,k^g=kl

Conjugacy classes & character table

448 conjugacy classes — view the full character table.

Relationships

automorphism group of

#11855, #11860, #11861, …

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