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C₅²×C₈ #6030

order	200
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×GL(2,𝔽₅) (order 1920) · Inn(G) = C₁ · Out(G) = C₂²×GL(2,𝔽₅)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₅

Aliases

C₅²×C₈

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂×C₅²
socle, prime radical	C₂×C₅²	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 200

A minimal faithful permutation representation — 2 generators on 18 points:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

center of	C₅²×M₃₂, C₅²×#131, C₅²×C₃⋊C₁₆, …
derived subgroup of	#11716586, #11716587, #11716588, …
Fitting subgroup of	C₅×#672, #87756, C₅×C₈×D₅, …
maximal subgroup of	C₅×#672, #87756, C₅²×C₁₆, …
maximal normal subgroup of	C₅×#672, #87756, C₅²×C₁₆, …
maximal quotient of	C₅²×C₁₆, C₂×C₅²×C₈, C₅²×C₃⋊C₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(200, 17);