Finite Group Databasebeta

← Prev (#672) Next (#674) →

C₅×C₁₆ #673

order	80
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₄² (order 32) · Inn(G) = C₁ · Out(G) = C₂×C₄²
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₅)×AGL(1,ℤ/16)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₈₀
abelian	C₅×C₁₆

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂×C₅
socle, prime radical	C₂×C₅	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

10 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 1 generator on 21 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

center of	C₅×M₆₄, C₅×#504, C₅×C₃⋊C₃₂, …
derived subgroup of	D₁₆₀, #64669, Dic₈₀, …
Frattini subgroup of	#11716342, C₂₅×C₃₂, #13097651, …
Fitting subgroup of	#3926, D₅×C₁₆, #3930, …
maximal subgroup of	#3926, C₅×C₃₂, D₅×C₁₆, …
maximal normal subgroup of	#3926, C₅×C₃₂, D₅×C₁₆, …
maximal quotient of	C₅×C₃₂, C₂×C₅×C₁₆, C₅×C₃⋊C₁₆, …
outer automorphism group of	C₁₇×D₁₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(80, 2);