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#13144250

order	1728
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

H(3,𝔽₃) ⋊ #431

Subgroups

	subgroup	G/H
center	C₃×C₄²	C₂×C₃⋊D₃
commutator	C₂×H(3,𝔽₃)	C₂²×C₈
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₄²×H(3,𝔽₃)	C₂
socle	C₂²×C₃	C₂²×C₃⋊Dic₃
prime radical	C₂³×H(3,𝔽₃)	C₂×C₄
cosocle	C₂×C₄×H(3,𝔽₃)	C₂³
hypercenter	C₂×C₃×C₄²	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#431
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃×C₄² ◁ C₂×C₃×C₄²

649 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 288

Polycyclic presentation:

2^6,3^3|a^2=e,b^2=d,e^2=f,c^a=cd,g^a=g^2,h^a=h^2,h^g=hi

Conjugacy classes & character table

248 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 10313);