#72340822

order	6144
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×#63011 ⋊ D₃
derived	C₂²⋊#5464 ⋊ C₂³

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₂²⋊#5464	C₂³
Frattini	C₂²	C₂²×#87588
Fitting	C₂²×#63011	D₃
socle	C₂⁴	C₂²×#1020
LCS	G ▷ C₂²⋊#5464
UCS	C₁

At least 16380 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has at least 16380 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,3,2^8|b^2=kl,i^2=l,j^2=k,d^b=d^2l,e^b=efk,e^c=el,e^d=f,f^b=fkl,f^c=fkl,f^d=ef,g^b=gh,g^d=h,h^d=gh,i^a=il,i^b=jkl,i^d=ij,i^e=il,i^f=ik,j^a=jk,j^b=ikl,j^d=ik,j^e=jkl,j^f=jl,k^b=l,k^d=l,l^b=k,l^d=kl

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340865
maximal normal subgroup of	#72340865

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