#72253276

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

Aliases

No aliases for this group.

Splittings

Fitting	H(3,𝔽₃)×H(5,𝔽₂) ⋊ C₂²
Sylow	H(3,𝔽₃) ⋊ C₂×B(4,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂	#13152097
commutator, cosocle	C₂³×H(3,𝔽₃)	C₂⁴
Frattini	C₂³×C₃	C₂²×D₃²
Fitting	H(3,𝔽₃)×H(5,𝔽₂)	C₂²
socle	C₂×C₃	#10595527
hypercenter	H(5,𝔽₂)	((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 ◁ H(5,𝔽₂)

4461 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 4461 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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^c=i^2,i^d=i^2j,i^h=ij,j^c=j^2

Conjugacy classes & character table

115 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, 3, 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, 3, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);