#72253277

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

Aliases

No aliases for this group.

Splittings

Fitting	H(3,𝔽₃)×#120 ⋊ C₂²
Sylow	H(3,𝔽₃) ⋊ C₂×B(4,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂	#13151503
commutator, cosocle	C₂³×H(3,𝔽₃)	C₂⁴
Frattini	C₂³×C₃	C₂²×D₃²
Fitting	H(3,𝔽₃)×#120	C₂²
socle	C₂×C₃	#10595278
hypercenter	#120	((C₃⋊D₃)⋊C₃)⋊C₂
hypocenter	H(3,𝔽₃)	C₂×B(4,𝔽₂)
LCS	G ▷ C₂³×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #120

4225 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 72

Polycyclic presentation:

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

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	B(4,𝔽₂)×((C₃⋊D₃)⋊C₃)⋊C₂
maximal normal subgroup of	B(4,𝔽₂)×((C₃⋊D₃)⋊C₃)⋊C₂

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