Q₁₆ #37

order	16
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = AGL(1,ℤ/8) (order 32) · Inn(G) = D₄ · Out(G) = C₂²
holomorph	G ⋊ Aut(G) = #489347
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄
quaternion	Q₁₆

	subgroup	G/H
center, socle, prime radical	C₂	D₄
commutator, Frattini, cosocle	C₄	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

9 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 16

A minimal faithful permutation representation — 2 generators on 16 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)

Polycyclic presentation:

2^4|a^2=d,b^2=d,c^2=d,b^a=bc,c^a=cd,c^b=cd

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16, 9);