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C₂²⋊C₄×#1084 #13144287

order	1728
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

C₂²⋊C₄×#1084

Splittings

H(3,𝔽₃) ⋊ C₄×C₂²⋊C₄

Subgroups

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

1419 conjugacy classes of subgroups; only the 301 normal subgroups are shown.

normal subgroups only

This group has 1419 conjugacy classes of subgroups — too many to draw. Only its 301 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) = 288

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 10350);