D₅×C₄₁ #88013

order	410
isoclinism	isoclinic class of D₅
automorphism	Aut(G) = C₅×C₈×AGL(1,𝔽₅) (order 800) · Inn(G) = D₅ · Out(G) = C₂×C₅×C₈
homology	Abelianization (H₁) = C₂×C₄₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₅×C₄₁
unambiguous semidirect product	C₅⋊(C₂×C₄₁)

Fitting, socle, Hall	C₅×C₄₁ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂×C₄₁

	subgroup	G/H
center, hypercenter	C₄₁	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₄₁
Frattini	C₁	G
Fitting, socle	C₅×C₄₁	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₄₁

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 46
transitive degree	μ_t(G) = 205

A minimal faithful permutation representation — 2 generators on 46 points:

(2,3)(4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46);(1,2,4,5,3)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(410, 3);