#72268985

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₃³×#102 ⋊ C₂²
Sylow	C₃³ ⋊ C₂×#442

Subgroups

	subgroup	G/H
center	C₂²	D₃×D₄×C₃⋊D₃
commutator	C₃³×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂²×D₃×C₃⋊D₃
Fitting	C₃³×#102	C₂²
socle	C₂²×C₃³	C₂²×D₄
prime radical	C₂×#11723547	C₂
cosocle	C₂×C₃³×C₄	C₂⁴
hypercenter	#102	D₃×C₃⋊D₃
hypocenter	C₃³	C₂×#442
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

4456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 288

Polycyclic presentation:

2^7,3^3|a^2=e,b^2=efg,c^2=f,d^2=f,f^2=g,b^a=bf,c^a=cf,c^b=cg,d^a=df,d^b=dg,f^a=fg,h^d=h^2,i^d=i^2,j^c=j^2

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×C₃⋊D₃×#442
maximal normal subgroup of	D₃×C₃⋊D₃×#442

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