#72258286

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃³×C₂²⋊C₄ ⋊ C₂³
Sylow	C₃³ ⋊ C₂×B(4,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂	#13171524
commutator, cosocle	C₂³×C₃³	C₂⁴
Frattini	C₂³	C₂×D₃³
Fitting	C₃³×C₂²⋊C₄	C₂³
socle	C₂×C₃³	C₂×#120
hypercenter	C₂²⋊C₄	D₃³
hypocenter	C₃³	C₂×B(4,𝔽₂)
LCS	G ▷ C₂³×C₃³ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²⋊C₄

5260 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 48

Polycyclic presentation:

2^7,3^3|a^2=g,c^a=cf,c^b=ce,e^a=eg,f^b=fg,h^a=h^2,h^d=h^2,i^a=h^2i,i^d=i^2,j^b=j^2,j^c=j^2

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

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