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D₄×((C₃⋊D₃)⋊C₃)⋊C₂ #11722503

order	864
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

D₄×((C₃⋊D₃)⋊C₃)⋊C₂

Splittings

Fitting	D₄×H(3,𝔽₃) ⋊ C₂²
Sylow	H(3,𝔽₃) ⋊ C₂²×D₄

Subgroups

	subgroup	G/H
center	C₂	C₂²×((C₃⋊D₃)⋊C₃)⋊C₂
commutator, cosocle	C₂×H(3,𝔽₃)	C₂⁴
Frattini, socle	C₂×C₃	C₂²×D₃²
Fitting	D₄×H(3,𝔽₃)	C₂²
hypercenter	D₄	((C₃⋊D₃)⋊C₃)⋊C₂
hypocenter	H(3,𝔽₃)	C₂²×D₄
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂ ◁ D₄

880 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 36

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#13146640, #13146642, D₈×((C₃⋊D₃)⋊C₃)⋊C₂, …
maximal normal subgroup of	#13146640, #13146642, D₈×((C₃⋊D₃)⋊C₃)⋊C₂, …
maximal quotient of	#13148455, #13148456, #13148457, …
automorphism group of	#634⋊C₃, C₄⋊((C₃⋊D₃)⋊C₃), C₂×(C₃⋊Dic₃)⋊C₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 2470);