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C₂×C₅ #18

order	10
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₄ (order 4) · Inn(G) = C₁ · Out(G) = C₄
holomorph	G ⋊ Aut(G) = C₂×AGL(1,𝔽₅)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₀
abelian	C₂×C₅
multiplicative	(ℤ/11)ˣ, (ℤ/22)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator, Frattini	C₁	G
LCS	G ▷ C₁
UCS	C₁ ◁ G

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 7
transitive degree	μ_t(G) = 10

A minimal faithful permutation representation — 1 generator on 7 points:

(1,2)(3,4,5,6,7)

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀
size	1	1	1	1	1	1	1	1	1	1
order	1	5	5	5	5	2	10	10	10	10
χ₁	1	1	1	1	1	1	1	1	1	1
χ₂	1	ζ	ζ²	ζ³	ζ⁴	1	ζ	ζ²	ζ³	ζ⁴
χ₃	1	ζ²	ζ⁴	ζ	ζ³	1	ζ²	ζ⁴	ζ	ζ³
χ₄	1	ζ³	ζ	ζ⁴	ζ²	1	ζ³	ζ	ζ⁴	ζ²
χ₅	1	ζ⁴	ζ³	ζ²	ζ	1	ζ⁴	ζ³	ζ²	ζ
χ₆	1	1	1	1	1	-1	-1	-1	-1	-1
χ₇	1	ζ	ζ²	ζ³	ζ⁴	-1	−ζ	−ζ²	−ζ³	−ζ⁴
χ₈	1	ζ²	ζ⁴	ζ	ζ³	-1	−ζ²	−ζ⁴	−ζ	−ζ³
χ₉	1	ζ³	ζ	ζ⁴	ζ²	-1	−ζ³	−ζ	−ζ⁴	−ζ²
χ₁₀	1	ζ⁴	ζ³	ζ²	ζ	-1	−ζ⁴	−ζ³	−ζ²	−ζ

Here ζ is a primitive root of unity of order 5 (ζ⁵ = 1).

Relationships

center of	C₅×D₄, C₅×Q₈, C₅×Dic₃, …
derived subgroup of	Dic₁₀, D₂₀, C₂²⋊D₅, …
Frattini subgroup of	Dic₂₅, C₄×C₂₅, #6016, …
Fitting subgroup of	Dic₅, C₂×D₅, #170, …
socle of	Dic₅, C₄×C₅, C₂×D₅, …
maximal subgroup of	Dic₅, C₄×C₅, C₂×D₅, …
maximal normal subgroup of	Dic₅, C₄×C₅, C₂×D₅, …
maximal quotient of	C₄×C₅, C₂²×C₅, C₅×D₃, …
automorphism group of	C₁₁, C₂×C₁₁
outer automorphism group of	C₁₁, C₂×C₁₁, Dic₁₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10, 2);