order	32
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = #23882 (order 256) · Inn(G) = D₄ · Out(G) = #120
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

C₂×Q₁₆

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²	D₄
commutator, Frattini, cosocle	C₄	C₂³
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

30 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 18 points:

(1,2);(3,4,7,10)(5,11,13,18)(6,14,16,9)(8,15,17,12);(3,5,7,13)(4,8,10,17)(6,15,16,12)(9,18,14,11)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32, 41);