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C₂×C₂₅ #254

order	50
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₄×C₅ (order 20) · Inn(G) = C₁ · Out(G) = C₄×C₅
holomorph	G ⋊ Aut(G) = C₂×AGL(1,ℤ/25)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₅₀
abelian	C₂×C₂₅

Subgroups

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

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 50

A minimal faithful permutation representation — 1 generator on 27 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)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

center of	D₄×C₂₅, Q₈×C₂₅, Dic₃×C₂₅, …
derived subgroup of	Dic₅₀, D₁₀₀, C₂²⋊D₂₅, …
Frattini subgroup of	Dic₁₂₅, C₄×C₁₂₅, #11758618, …
Fitting subgroup of	Dic₂₅, C₂×D₂₅, #6016, …
maximal subgroup of	Dic₂₅, C₄×C₂₅, C₂×D₂₅, …
maximal normal subgroup of	Dic₂₅, C₄×C₂₅, C₂×D₂₅, …
maximal quotient of	C₄×C₂₅, C₂²×C₂₅, D₃×C₂₅, …
outer automorphism group of	D₁₀₁, D₁₂₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(50, 2);