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C₃×C₁₆ #200

order	48
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₄ (order 16) · Inn(G) = C₁ · Out(G) = C₂²×C₄
holomorph	G ⋊ Aut(G) = D₃×AGL(1,ℤ/16)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₄₈
abelian	C₃×C₁₆

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂×C₃
socle, prime radical	C₂×C₃	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

10 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 1 generator on 19 points:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

center of	C₃×M₆₄, C₃×#504, C₃×C₃⋊C₃₂, …
derived subgroup of	D₉₆, #4449, Dic₄₈, …
Frattini subgroup of	C₉⋊C₃₂, C₉×C₃₂, D₉×C₃₂, …
Fitting subgroup of	C₃⋊C₃₂, D₃×C₁₆, #798, …
maximal subgroup of	C₃⋊C₃₂, C₃×C₃₂, D₃×C₁₆, …
maximal normal subgroup of	C₃⋊C₃₂, C₃×C₃₂, D₃×C₁₆, …
maximal quotient of	C₃×C₃₂, C₂×C₃×C₁₆, C₉×C₁₆, …
outer automorphism group of	D₉₇, D₇×C₁₇, D₁₁₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(48, 2);