#72269595

order	4608
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×#12045 ⋊ C₂
Sylow	C₃² ⋊ #151085

Subgroups

	subgroup	G/H
center	C₂²	#11797804
commutator	C₃²×C₁₆	C₂²×C₈
Frattini	C₄×C₁₆	C₂²×C₃⋊D₃
Fitting	C₃²×#12045	C₂
socle	C₂²×C₃²	#2177
prime radical	C₂×#10589385	C₄
cosocle	C₃²×C₄×C₁₆	C₂³
hypercenter	#12045	C₃⋊D₃
hypocenter	C₃²	#151085
LCS	G ▷ C₃²×C₁₆ ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311 ◁ #12045

924 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 2304

Polycyclic presentation:

2^9,3^2|a^2=g,b^2=dfi,c^2=e,d^2=fh,e^2=g,f^2=hi,h^2=i,b^a=bd,c^b=ch,d^a=df,d^c=di,e^b=ei,f^a=fh,h^a=hi,j^b=j^2,k^b=k^2

Conjugacy classes & character table

324 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃⋊D₃×#151085
maximal normal subgroup of	C₃⋊D₃×#151085

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