Finite Group Databasebeta

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D₃×#12696 #72253545

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

Splittings

Fitting	C₃×#12696 ⋊ C₂
Sylow	C₃ ⋊ C₂×#12696

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₃×#99
commutator	C₂²×C₃×C₄	C₂³×C₄
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₃×#12696	C₂
socle	C₂²×C₃	C₂²×#99
prime radical	C₂×D₃×#535	C₂
cosocle	C₂²×C₃×D₄	C₂⁴
hypercenter	#12696	D₃
hypocenter	C₃	C₂×#12696
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #12696

8524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#493370, D₃×#133356, D₃×#499832, …
maximal normal subgroup of	D₃×#493370, D₃×#133356, D₃×#499832, …

GAP

Past the SmallGroups library coverage — no SmallGroups id.