#72219779

order	4608
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×#1455 ⋊ C₂²
Sylow	C₃² ⋊ #489347

Subgroups

	subgroup	G/H
center	C₂	#72219778
commutator, cosocle	C₃²×C₄×C₈	C₂⁴
Frattini	C₄×C₈	C₂²×D₃²
Fitting	C₃²×#1455	C₂²
socle	C₂×C₃²	#18524
hypercenter	#1455	D₃²
hypocenter	C₃²	#489347
LCS	G ▷ C₃²×C₄×C₈ ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #1455

4337 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^9,3^2|a^2=i,b^2=e,e^2=g,f^2=h,h^2=i,b^a=bf,c^b=ce,d^b=dg,e^a=eh,e^c=eg,f^a=fhi,f^c=fhi,f^d=fi,g^a=gi,h^a=hi,h^c=hi,j^a=j^2,j^c=j^2,k^a=j^2k,k^c=k^2

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal normal subgroup of

#72219802

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