C₄⋊C₄ #32

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) = #346446
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

C₄⋊C₄

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

13 conjugacy classes of subgroups.

minimal degree	μ(G) = 8
transitive degree	μ_t(G) = 16

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

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

Polycyclic presentation:

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16, 4);