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D₃×#1728 #10792539

order	768
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

Splittings

Fitting	C₃×#1728 ⋊ C₂
Sylow	C₃ ⋊ C₂×#1728

Subgroups

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₃
commutator	C₂²×C₃	C₂²×C₄²
Fitting	C₃×#1728	C₂
socle	C₂³×C₃	C₂³×C₄
prime radical	C₂⁴×D₃	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂⁴
hypercenter	#1728	D₃
hypocenter	C₃	C₂×#1728
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1728

2782 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 192

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#10746, D₃×#23430, D₃×#11041
maximal normal subgroup of	D₃×#10746, D₃×#23430, D₃×#11041

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 166920);