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D₃×#101 #4750

order	192
automorphism	Aut(G) = C₂×D₃×#2196 (order 1536) · Inn(G) = D₃×C₂²⋊C₄ · Out(G) = C₂×D₄
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

Fitting	C₃×#101 ⋊ C₂
Sylow	C₃ ⋊ C₂×#101

Subgroups

	subgroup	G/H
center	C₂	D₃×C₂²⋊C₄
commutator	C₂²×C₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×#101	C₂
socle	C₂×C₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	#101	D₃
hypocenter	C₃	C₂×#101
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

146 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)(4,6,8,14)(5,9,10,12)(7,15,16,13)(11,19,18,17);(4,5,7,11,8,10,16,18)(6,12,13,19,14,9,15,17);(2,3)

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#355, #69164, D₃×#356, …
maximal normal subgroup of	D₃×#355, #69164, D₃×#356, …
maximal quotient of	#68125, #68127, #68128, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 309);