#72271449

order	6144
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting

#72268062 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²	#72209718
commutator, cosocle	C₂×#3459⋊C₃	C₂³
Frattini	C₂²×C₄	C₂²×#1020
Fitting	#72268062	D₃
hypercenter	#534	#1020
hypocenter	#543⋊C₃	#120
LCS	G ▷ C₂×#3459⋊C₃ ▷ #543⋊C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #534

Only the 114 normal subgroups are shown; non-normal subgroups are omitted.

normal subgroups only

Only the group's 114 normal subgroups are shown; non-normal subgroups are omitted.

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

Full subgroup data →

Presentation

Polycyclic presentation:

2^5,3,2^6|e^2=kl,i^2=k,j^2=k,b^a=bl,c^a=cd,c^b=ce,d^b=dl,e^a=el,e^b=ekl,e^c=ekl,f^c=f^2,g^c=h,g^f=h,h^c=g,h^f=gh,i^c=ghikl,i^f=j,i^g=il,i^h=il,j^c=hijl,j^f=ijk,j^h=jl,j^i=jk

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72271469

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