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C₄×C₁₃ #260

order	52
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₃×C₄ (order 24) · Inn(G) = C₁ · Out(G) = C₂×C₃×C₄
holomorph	G ⋊ Aut(G) = D₄×AGL(1,𝔽₁₃)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₅₂
abelian	C₄×C₁₃
multiplicative	(ℤ/53)ˣ, (ℤ/106)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂×C₁₃
socle, prime radical	C₂×C₁₃	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 52

A minimal faithful permutation representation — 1 generator on 17 points:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

center of	C₁₃×M₁₆, C₁₃×#41, C₁₃×C₃⋊C₈, …
derived subgroup of	#6095, D₁₀₄, Dic₅₂, …
Frattini subgroup of	#13062555, C₈×C₁₆₉, D₁₄₈₇₂, …
Fitting subgroup of	#1055, Dic₂₆, C₄×D₁₃, …
maximal subgroup of	#1055, C₈×C₁₃, Dic₂₆, …
maximal normal subgroup of	#1055, C₈×C₁₃, Dic₂₆, …
maximal quotient of	C₈×C₁₃, C₂×C₄×C₁₃, Dic₃×C₁₃, …
automorphism group of	C₅₃, C₂×C₅₃
outer automorphism group of	C₅₃, C₂×C₅₃, D₃×C₅₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(52, 2);