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C₂×C₅² #257

order	50
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = GL(2,𝔽₅) (order 480) · Inn(G) = C₁ · Out(G) = GL(2,𝔽₅)
holomorph	G ⋊ Aut(G) = C₂×AGL(2,𝔽₅)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₅

Aliases

C₂×C₅²

Subgroups

	subgroup	G/H
center	G	C₁
commutator, Frattini	C₁	G
LCS	G ▷ C₁
UCS	C₁ ◁ G

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 50

A minimal faithful permutation representation — 2 generators on 12 points:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

center of	C₅²×D₄, C₅²×Q₈, C₅²×Dic₃, …
derived subgroup of	#6037, #6038, #6039, …
Fitting subgroup of	C₅×Dic₅, #1039, C₂×C₅×D₅, …
socle of	C₅×Dic₅, #1039, C₄×C₅², …
maximal subgroup of	C₅×Dic₅, #1039, C₄×C₅², …
maximal normal subgroup of	C₅×Dic₅, #1039, C₄×C₅², …
maximal quotient of	C₄×C₅², C₂²×C₅², C₅²×D₃, …
outer automorphism group of	C₁₁×D₁₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(50, 5);