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D₃×SD₃₂ #4913

order	192
automorphism	Aut(G) = C₂×D₃×#442 (order 768) · Inn(G) = D₃×D₈ · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

D₃×SD₃₂

Splittings

Fitting	C₃×SD₃₂ ⋊ C₂
Sylow	C₃ ⋊ C₂×SD₃₂

Subgroups

	subgroup	G/H
center	C₂	D₃×D₈
commutator, cosocle	C₃×C₈	C₂³
Frattini	C₈	C₂²×D₃
Fitting	C₃×SD₃₂	C₂
socle	C₂×C₃	C₂×D₈
prime radical	D₃×D₈	C₂
hypercenter	SD₃₂	D₃
hypocenter	C₃	C₂×SD₃₂
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ SD₃₂

90 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:

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

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×D₃×SD₃₂, #82024, D₃×#508, …
maximal normal subgroup of	C₂×D₃×SD₃₂, #82024, D₃×#508, …
maximal quotient of	#69175, #69179, #69183, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 472);