C₄₁⋊C₅ #6084

order	205
isoclinism	isoclinic class of C₄₁⋊C₅
automorphism	Aut(G) = AGL(1,𝔽₄₁) (order 1640) · Inn(G) = G · Out(G) = C₈
homology	Abelianization (H₁) = C₅ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₄₁⋊C₅

Splittings

Fitting, socle, derived, Sylow

C₄₁ ⋊ C₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₄₁	C₅
LCS	G ▷ C₄₁
UCS	C₁

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 41 points:

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

Polycyclic presentation:

5,41|b^a=b^10

Conjugacy classes & character table

13 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄₁⋊C₅, C₂×C₄₁⋊C₅, C₃×C₄₁⋊C₅, …
maximal normal subgroup of	D₄₁⋊C₅, C₂×C₄₁⋊C₅, C₃×C₄₁⋊C₅, …
maximal quotient of	C₂×C₄₁⋊C₅, C₃×C₄₁⋊C₅, C₄₁⋊C₂₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(205, 1);

C41⋊C5 #6084