#72244376

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	3₋¹⁺²×#361 ⋊ C₂
Sylow	3₋¹⁺² ⋊ AGL(1,ℤ/16)

Subgroups

	subgroup	G/H
center	C₂	#10588465⋊C₃
commutator	C₈×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃×C₈	C₂²×C₃×D₃
Fitting	3₋¹⁺²×#361	C₂
socle	C₂×C₃	C₃×#4785
prime radical	#10588827⋊C₃	C₂
cosocle	C₂×C₈×C₉	C₂³×C₃
hypercenter	#361	AGL(1,ℤ/9)
hypocenter	C₉	C₃×AGL(1,ℤ/16)
LCS	G ▷ C₈×C₉ ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #361

493 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 144

Polycyclic presentation:

2^3,3,2^4,3^2|a^2=f,b^2=e,c^2=e,f^2=g,g^2=h,i^3=j,b^a=bfgh,c^a=cgh,e^a=eh,f^b=fg,f^c=fh,g^b=gh,i^a=i^2j^2,i^b=i^2j^2,i^c=i^2j^2,i^d=ij,j^a=j^2,j^b=j^2,j^c=j^2

Conjugacy classes & character table

133 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/9)×AGL(1,ℤ/16)
maximal normal subgroup of	AGL(1,ℤ/9)×AGL(1,ℤ/16)

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