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D₃×#106 #4859

order	192
automorphism	Aut(G) = D₃×#6628842 (order 3072) · Inn(G) = D₃×D₄ · Out(G) = C₂×#120
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

Fitting	C₃×#106 ⋊ C₂
Sylow	C₃ ⋊ C₂×#106

Subgroups

	subgroup	G/H
center	C₂²	D₃×D₄
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×#106	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	C₂²×D₃	D₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	#106	D₃
hypocenter	C₃	C₂×#106
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#360, D₃×#365, C₂×D₃×#106, …
maximal normal subgroup of	D₃×#360, D₃×#365, C₂×D₃×#106, …
maximal quotient of	D₃×#334, #68529, #68532, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 418);