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C₅×C₈ #169

order	40
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) = AGL(1,𝔽₅)×AGL(1,ℤ/8)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₄₀
abelian	C₅×C₈
multiplicative	(ℤ/41)ˣ, (ℤ/82)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂×C₅
socle, prime radical	C₂×C₅	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 40

A minimal faithful permutation representation — 1 generator on 13 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

center of	C₅×M₃₂, C₅×#131, C₅×C₃⋊C₁₆, …
derived subgroup of	D₈₀, #3932, Dic₄₀, …
Frattini subgroup of	#87707, C₁₆×C₂₅, #11716344, …
Fitting subgroup of	#672, C₈×D₅, #676, …
maximal subgroup of	#672, C₅×C₁₆, C₈×D₅, …
maximal normal subgroup of	#672, C₅×C₁₆, C₈×D₅, …
maximal quotient of	C₅×C₁₆, C₂×C₅×C₈, C₅×C₃⋊C₈, …
automorphism group of	C₄₁, C₂×C₄₁
outer automorphism group of	C₄₁, C₂×C₄₁, D₃×C₄₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(40, 2);