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C₂×C₄×((C₃⋊D₃)⋊C₃)⋊C₄ #13150753

order	1728
isoclinism	isoclinic class of ((C₃⋊D₃)⋊C₃)⋊C₂
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

C₂×C₄×((C₃⋊D₃)⋊C₃)⋊C₄

Splittings

H(3,𝔽₃) ⋊ C₂²×C₄²

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₄	((C₃⋊D₃)⋊C₃)⋊C₂
commutator, hypocenter	H(3,𝔽₃)	C₂²×C₄²
Frattini	C₂²×C₃	C₂²×D₃²
Fitting	C₂²×C₄×H(3,𝔽₃)	C₂²
socle	C₂³×C₃	C₂×D₃²
prime radical	C₂³×(C₃⋊D₃)⋊C₃	C₂²
cosocle	C₂²×H(3,𝔽₃)	C₂⁴
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₄

1851 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 288

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 16816);