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M₁₆×((C₃⋊D₃)⋊C₃)⋊C₂ #13149874

order	1728
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

M₁₆×((C₃⋊D₃)⋊C₃)⋊C₂

Splittings

Fitting	M₁₆×H(3,𝔽₃) ⋊ C₂²
Sylow	H(3,𝔽₃) ⋊ C₂²×M₁₆

Subgroups

	subgroup	G/H
center	C₄	C₂²×((C₃⋊D₃)⋊C₃)⋊C₂
commutator	C₂×H(3,𝔽₃)	C₂³×C₄
Frattini	C₃×C₄	C₂²×D₃²
Fitting	M₁₆×H(3,𝔽₃)	C₂²
socle	C₂×C₃	C₂×C₄×D₃²
prime radical	C₂²×((C₃⋊D₃)⋊C₃)⋊C₂	C₄
cosocle	C₄×H(3,𝔽₃)	C₂⁴
hypercenter	M₁₆	((C₃⋊D₃)⋊C₃)⋊C₂
hypocenter	H(3,𝔽₃)	C₂²×M₁₆
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₄ ◁ M₁₆

1111 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 72

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	M₁₆×AGL(2,𝔽₃), AGL(1,ℤ/8)×((C₃⋊D₃)⋊C₃)⋊C₂
maximal normal subgroup of	AGL(1,ℤ/8)×((C₃⋊D₃)⋊C₃)⋊C₂
maximal quotient of	#97×((C₃⋊D₃)⋊C₃)⋊C₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 15937);