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D₃×#107 #4879

order	192
automorphism	Aut(G) = C₂²×D₃×#3003 (order 3072) · Inn(G) = D₃×D₄ · Out(G) = C₂³×D₄
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

Fitting	C₃×#107 ⋊ C₂
Sylow	C₃ ⋊ C₂×#107

Subgroups

	subgroup	G/H
center	C₂²	D₃×D₄
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×#107	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	C₂²×D₃	D₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	#107	D₃
hypocenter	C₃	C₂×#107
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

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:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#357, #69189, #69199, …
maximal normal subgroup of	D₃×#357, #69189, #69199, …
maximal quotient of	D₃×#335, #68569, #68572, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 438);