#72252386

order	4608
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃²×B(4,𝔽₂) ⋊ C₂²
Sylow	C₃² ⋊ #511790

Subgroups

	subgroup	G/H
center	C₂	#72252270
commutator, cosocle	C₂²×C₃²×D₄	C₂⁴
Frattini	C₂²×D₄	C₂²×D₃²
Fitting	C₂×C₃²×B(4,𝔽₂)	C₂²
socle	C₂×C₃²	#32898
hypercenter	C₂×B(4,𝔽₂)	D₃²
hypocenter	C₃²	#511790
LCS	G ▷ C₂²×C₃²×D₄ ▷ C₂²×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×D₄ ◁ C₂×B(4,𝔽₂)

At least 6293 conjugacy classes of subgroups; only the 358 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^9,3^2|a^2=hi,d^2=ef,g^2=i,b^a=bg,c^b=ce,d^a=dg,d^b=def,d^c=de,e^a=eh,f^a=fhi,g^b=gi,g^c=ghi,g^e=gi,g^f=gi,h^b=hi,h^d=hi,j^a=j^2,j^d=j^2,k^a=j^2k,k^d=k^2

Conjugacy classes & character table

153 conjugacy classes — view the full character table.

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