#72222660

order	9216
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×#12067 ⋊ C₂²
Sylow	C₃² ⋊ C₂×AGL(1,ℤ/32)

Subgroups

	subgroup	G/H
center	C₂	#72222659
commutator	C₃²×C₁₆	C₂³×C₈
Frattini	#346	C₂²×D₃²
Fitting	C₃²×#12067	C₂²
socle	C₂×C₃²	C₂×#12067
prime radical	#72222630	C₄
cosocle	C₃²×#346	C₂⁴
hypercenter	#12067	D₃²
hypocenter	C₃²	C₂×AGL(1,ℤ/32)
LCS	G ▷ C₃²×C₁₆ ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1313 ◁ #12067

3174 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^10,3^2|a^2=f,b^2=e,c^2=eg,e^2=g,f^2=h,h^2=i,i^2=j,b^a=bfhij,c^a=chij,e^a=eij,f^b=fh,f^c=fi,f^e=fj,g^a=gj,h^b=hi,h^c=hj,i^b=ij,k^a=k^2,k^d=k^2,l^a=k^2l,l^d=l^2

Conjugacy classes & character table

234 conjugacy classes — view the full character table.

Relationships

holomorph of

C₃⋊C₃₂

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