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D₃×#444 #79896

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

Aliases

Splittings

Fitting	C₃×#444 ⋊ C₂
Sylow	C₃ ⋊ C₂×#444

Subgroups

	subgroup	G/H
center	C₄	C₂×D₃×D₄
commutator	C₃×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#444	C₂
socle	C₂×C₃	C₂×C₄×D₄
prime radical	D₃×#135	C₂
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#444	D₃
hypocenter	C₃	C₂×#444
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #444

464 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#2180, D₃×AGL(1,ℤ/16), #10948632, …
maximal normal subgroup of	D₃×#2180, D₃×AGL(1,ℤ/16), #10948632, …
maximal quotient of	D₃×SD₁₆⋊C₈, D₃×#1579, D₃×#1580, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 12472);