#72219797

order	4608
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×#3184 ⋊ C₂²
Sylow	C₃² ⋊ C₂×#33688

Subgroups

	subgroup	G/H
center	C₂	#72219796
commutator, cosocle	C₂×C₃²×C₈	C₂⁵
Frattini	C₂×C₈	C₂³×D₃²
Fitting	C₃²×#3184	C₂²
socle	C₂×C₃²	C₂×#3109
hypercenter	#3184	D₃²
hypocenter	C₃²	C₂×#33688
LCS	G ▷ C₂×C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #3184

7773 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^9,3^2|a^2=i,e^2=g,f^2=hi,h^2=i,b^a=bf,c^b=cg,e^a=ehi,e^b=eg,f^a=fh,f^b=fh,f^c=fi,g^a=gi,h^a=hi,h^b=hi,j^a=j^2,j^d=j^2,k^a=j^2k,k^d=k^2

Conjugacy classes & character table

168 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, 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.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, 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, 2, 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);