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D₃×#344 #69025

order	384
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

Splittings

Fitting	C₃×#344 ⋊ C₂
Sylow	C₃ ⋊ C₂×#344

Subgroups

	subgroup	G/H
center	C₄	D₃×C₂²⋊C₄
commutator	C₂²×C₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#344	C₂
socle	C₂×C₃	C₂×#95
prime radical	C₂³×D₃	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#344	D₃
hypocenter	C₃	C₂×#344
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

388 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 48

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

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×D₃×#344, #10792642, D₃×#1757, …
maximal normal subgroup of	C₂×D₃×#344, #10792642, D₃×#1757, …
maximal quotient of	D₃×#1281, #10684816, #10684819, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 1601);