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C₄₁ #182

order	41
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₅×C₈ (order 40) · Inn(G) = C₁ · Out(G) = C₅×C₈
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₄₁)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

C₄₁

Subgroups

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

2 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 41
transitive degree	μ_t(G) = 41

A minimal faithful permutation representation — 1 generator on 41 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

center of	D₃×C₄₁, D₅×C₄₁, A₄×C₄₁, …
derived subgroup of	D₄₁, Dic₄₁, #4223, …
Frattini subgroup of	C₁₆₈₁, D₁₆₈₁, C₂×C₁₆₈₁, …
Fitting subgroup of	D₄₁, #4223, C₄₁⋊C₅, …
socle of	D₄₁, #4223, C₄₁⋊C₅, …
maximal subgroup of	D₄₁, C₂×C₄₁, C₃×C₄₁, …
maximal normal subgroup of	D₄₁, C₂×C₄₁, C₃×C₄₁, …
maximal quotient of	C₂×C₄₁, C₃×C₄₁, C₅×C₄₁, …
outer automorphism group of	D₈₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(41, 1);