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C₄×C₅² #1040

order	100
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×GL(2,𝔽₅) (order 960) · Inn(G) = C₁ · Out(G) = C₂×GL(2,𝔽₅)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₅

Aliases

C₄×C₅²

Subgroups

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

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 100

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

center of	C₅²×M₁₆, C₅²×#41, C₅²×C₃⋊C₈, …
derived subgroup of	#87770, #87771, #87772, …
Fitting subgroup of	C₅×#168, #6029, C₅×Dic₁₀, …
maximal subgroup of	C₅×#168, #6029, C₅²×C₈, …
maximal normal subgroup of	C₅×#168, #6029, C₅²×C₈, …
maximal quotient of	C₅²×C₈, C₂×C₄×C₅², C₅²×Dic₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(100, 8);