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D₃×#440 #79771

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

Aliases

Splittings

Fitting	C₃×#440 ⋊ C₂
Sylow	C₃ ⋊ C₂×#440

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator	C₃×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#440	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×D₃×D₄	C₂²
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#440	D₃
hypocenter	C₃	C₂×#440
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

544 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 15 points:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×D₃×#440, D₃×#2943, C₄×D₃×AGL(1,ℤ/8), …
maximal normal subgroup of	C₂×D₃×#440, D₃×#2943, C₄×D₃×AGL(1,ℤ/8), …
maximal quotient of	D₃×SD₁₆⋊C₈, D₃×#1585, D₃×#1588, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 12347);