Dic₁₀ #171

order	40
isoclinism	isoclinic class of Dic₁₀
automorphism	Aut(G) = D₄×AGL(1,𝔽₅) (order 160) · Inn(G) = C₂×D₅ · Out(G) = C₂×C₄
holomorph	G ⋊ Aut(G) = #72213584
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₁₀
unambiguous semidirect product	C₅⋊Q₈

C₅ ⋊ Q₈

	subgroup	G/H
center, Frattini	C₂	C₂×D₅
commutator, socle, prime radical, cosocle	C₂×C₅	C₂²
Fitting	C₄×C₅	C₂
hypercenter	C₄	D₅
hypocenter	C₅	Q₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₄

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 40

A minimal faithful permutation representation — 2 generators on 13 points:

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

Polycyclic presentation:

2^3,5|a^2=c,b^2=c,b^a=bc,d^a=d^4

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(40, 4);