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D₃×#108 #4892

order	192
automorphism	Aut(G) = C₂²×D₃×AGL(1,ℤ/8) (order 768) · Inn(G) = D₃×D₄ · Out(G) = C₂⁴
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

Fitting	C₃×#108 ⋊ C₂
Sylow	C₃ ⋊ C₂×#108

Subgroups

	subgroup	G/H
center	C₄	D₃×D₄
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×#108	C₂
socle	C₂×C₃	C₂×C₄⋊C₄
prime radical	C₂²×D₃	Q₈
cosocle	C₂×C₃×C₄	C₂³
hypercenter	#108	D₃
hypocenter	C₃	C₂×#108
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

106 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:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#359, #69241, D₃×#361, …
maximal normal subgroup of	D₃×#359, #69241, D₃×#361, …
maximal quotient of	D₃×#334, #68527, #68529, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 451);