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C₄×C₂₅ #1034

order	100
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₄×C₅ (order 40) · Inn(G) = C₁ · Out(G) = C₂×C₄×C₅
holomorph	G ⋊ Aut(G) = D₄×AGL(1,ℤ/25)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₀₀
abelian	C₄×C₂₅
multiplicative	(ℤ/101)ˣ, (ℤ/125)ˣ, (ℤ/202)ˣ, (ℤ/250)ˣ

Subgroups

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

9 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 100

A minimal faithful permutation representation — 1 generator on 29 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)

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

center of	M₁₆×C₂₅, #41×C₂₅, C₃⋊C₈×C₂₅, …
derived subgroup of	Dic₁₀₀, #87713, D₂₀₀, …
Frattini subgroup of	#11758616, C₈×C₁₂₅, #14522040, …
Fitting subgroup of	#6014, Dic₅₀, C₄×D₂₅, …
maximal subgroup of	#6014, C₈×C₂₅, Dic₅₀, …
maximal normal subgroup of	#6014, C₈×C₂₅, Dic₅₀, …
maximal quotient of	C₈×C₂₅, C₂×C₄×C₂₅, Dic₃×C₂₅, …
automorphism group of	C₁₀₁, C₁₂₅, C₂×C₁₀₁, …
outer automorphism group of	C₁₀₁, C₁₂₅, C₂×C₁₀₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(100, 2);