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C₂×#11720520 #13144246

order	1728
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

C₂×#11720520

Splittings

H(3,𝔽₃) ⋊ C₂×#97

Subgroups

	subgroup	G/H
center	C₂²×C₃×C₄	C₂×C₃⋊D₃
commutator	C₂×H(3,𝔽₃)	C₂×C₄²
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₄²×H(3,𝔽₃)	C₂
socle	C₂³×C₃	C₂×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)	C₂×#97
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄²

803 conjugacy classes of subgroups; only the 293 normal subgroups are shown.

normal subgroups only

This group has 803 conjugacy classes of subgroups — too many to draw. Only its 293 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) = 576

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, 10309);