D₄×C₄₁ #66500

order	328
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₅×C₈×D₄ (order 320) · Inn(G) = C₂² · Out(G) = C₂×C₅×C₈
homology	Abelianization (H₁) = C₂²×C₄₁ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₄×C₄₁
unambiguous semidirect product	C₄⋊(C₂×C₄₁)
unambiguous semidirect product	C₂²⋊(C₂×C₄₁)

	subgroup	G/H
center, socle	C₂×C₄₁	C₂²
commutator, Frattini, cosocle	C₂	C₂²×C₄₁
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄₁ ◁ G

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 45
transitive degree	μ_t(G) = 164

A minimal faithful permutation representation — 2 generators on 45 points:

(1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45);(2,4)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(328, 10);