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Dic₃×#123 #82835

order	384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

Splittings

C₃ ⋊ C₄×#123

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄
Fitting	C₂×C₃×#123	C₂
socle, cosocle	C₂³×C₃	C₂⁴
prime radical	C₂²×C₃×D₄	C₂²
hypercenter	C₂×#123	D₃
hypocenter	C₃	C₄×#123
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×#123

688 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₃×#352, #10711153, Dic₃×#458, …
maximal normal subgroup of	Dic₃×#352, #10711153, Dic₃×#458, …
maximal quotient of	Dic₃×#386, Dic₃×#388, Dic₃×#385, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 15411);