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D₃×#322 #68066

order	384
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

Splittings

Fitting	C₃×#322 ⋊ C₂
Sylow	C₃ ⋊ C₂×#322

Subgroups

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₃
commutator	C₂×C₃	C₂×C₄×C₈
Fitting	C₃×#322	C₂
socle	C₂²×C₃	C₂×C₄²
prime radical	C₂²×D₃	C₄²
cosocle	C₃×C₄²	C₂³
hypercenter	#322	D₃
hypocenter	C₃	C₂×#322
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄² ◁ #322

248 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 19 points:

(1,2,3,5,4,6,7,8)(10,13)(14,16)(17,18,19);(9,10,11,14,12,13,15,16);(18,19)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#1313, #10704186, #10704238, …
maximal normal subgroup of	D₃×#1313, #10704186, #10704238, …
maximal quotient of	Dic₃×#322, D₃×#1273, #10685609, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 642);