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C₄×#170 #4000

order	160
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	Aut(G) = AGL(1,𝔽₅)×#120 (order 640) · Inn(G) = AGL(1,𝔽₅) · Out(G) = #120
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₄

Aliases

Splittings

C₅ ⋊ C₄×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₄×C₈
Frattini	C₂²	C₂×AGL(1,𝔽₅)
Fitting	C₂×C₄×C₅	C₄
socle, prime radical	C₂²×C₅	C₂×C₄
cosocle	C₂×Dic₅	C₂²
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₄

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 2 generators on 17 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#64871, #64872, C₈×#170, …
maximal normal subgroup of	#64871, #64872, C₈×#170, …
maximal quotient of	C₄×#674, #64859, C₈×#170, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 75);