#72220980

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₃³×#120 ⋊ C₂²
Sylow	C₃³ ⋊ C₂×B(4,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂	#13178702
commutator, cosocle	C₂³×C₃³	C₂⁴
Frattini	C₂³	C₂²×#1113
Fitting	C₃³×#120	C₂²
socle	C₂×C₃³	C₂×#120
hypercenter	#120	#1113
hypocenter	C₃³	C₂×B(4,𝔽₂)
LCS	G ▷ C₂³×C₃³ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #120

5260 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^7,3^3|d^a=df,d^b=de,e^a=eg,f^b=fg,h^a=h^2,h^c=h^2,i^a=h^2i,i^c=i^2,j^a=j^2

Conjugacy classes & character table

162 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220991, D₃×#11865254
maximal normal subgroup of	D₃×#11865254

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