D₄×Q₈ #549

order	64
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	Aut(G) = #120×#1020 (order 3072) · Inn(G) = C₂⁴ · Out(G) = D₄×S₄
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

D₄×Q₈

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁴
prime radical	C₂×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

140 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 12 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 230);