order	16
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = #120 (order 32) · Inn(G) = C₂² · Out(G) = D₄
holomorph	G ⋊ Aut(G) = #511790
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

C₂²⋊C₄

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂²	C₂²
commutator	C₂	C₂×C₄
prime radical	C₂³	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

17 conjugacy classes of subgroups.

minimal degree	μ(G) = 8
transitive degree	μ_t(G) = 8

A minimal faithful permutation representation — 2 generators on 8 points:

(1,2,4,6)(3,5,7,8);(2,5)(6,8)

Polycyclic presentation:

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

Here i is the imaginary unit (i² = −1).

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16, 3);