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Dic₃×C₁₆ #4500

order	192
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂×D₃×#525 (order 768) · Inn(G) = D₃ · Out(G) = C₂×#525
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₄

Aliases

Dic₃×C₁₆

Splittings

C₃ ⋊ C₄×C₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₁₆	D₃
commutator, hypocenter	C₃	C₄×C₁₆
Frattini	C₂×C₈	C₂×D₃
Fitting	C₂×C₃×C₁₆	C₂
socle, prime radical	C₂²×C₃	C₂×C₈
cosocle	C₂×C₃×C₈	C₂²
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₁₆

58 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 2 generators on 23 points:

(2,3)(4,5,6,7);(1,2,3)(8,9,10,13,11,14,16,19,12,15,17,20,18,21,22,23)

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₃×C₃₂, #67589, Dic₃⋊C₃₂, …
maximal normal subgroup of	Dic₃×C₃₂, #67589, Dic₃⋊C₃₂, …
maximal quotient of	C₁₆×C₃⋊C₈, #67499, Dic₃×C₃₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 59);