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C₅×C₃₂ #3927

order	160
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₄×C₈ (order 64) · Inn(G) = C₁ · Out(G) = C₂×C₄×C₈
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₅)×AGL(1,ℤ/32)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₆₀
abelian	C₅×C₃₂

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₁₆	C₂×C₅
socle, prime radical	C₂×C₅	C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 1 generator on 37 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

center of	C₅×M₁₂₈, C₅×#2258, C₅×C₃⋊C₆₄, …
derived subgroup of	D₃₂₀, #10597925, Dic₁₆₀, …
Frattini subgroup of	#13097649, C₂₅×C₆₄, Dic₈₈₀₀, …
Fitting subgroup of	#64663, D₅×C₃₂, #64667, …
maximal subgroup of	#64663, C₅×C₆₄, D₅×C₃₂, …
maximal normal subgroup of	#64663, C₅×C₆₄, D₅×C₃₂, …
maximal quotient of	C₅×C₆₄, C₂×C₅×C₃₂, C₅×C₃⋊C₃₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 2);