▁For ▁even ▁n ▁we ▁have : ▁E +( n ) ▁/ ▁di h ▁( R n ) ▁ ▁SO ( n ) ▁/ ▁{ ▁I , ▁− I ▁} ▁ ▁Rot ation ▁In ▁ 3 D , ▁the ▁conjug ate ▁by ▁a ▁translation ▁of ▁a ▁rotation ▁about ▁an ▁axis ▁is ▁the ▁corresponding ▁rotation ▁about ▁the ▁translated ▁axis . ▁Such ▁a ▁conjug ation ▁produces ▁he ▁sc rew ▁disp lacement ▁known ▁to ▁express ▁an ▁arbitrary ▁E uclidean ▁motion ▁according ▁to ▁Ch as les ' ▁theorem . ▁ ▁The ▁conjug acy ▁class ▁within ▁the ▁E uclidean ▁group ▁E ( 3 ) ▁of ▁a ▁rotation ▁about ▁an ▁axis ▁is ▁a ▁rotation ▁by ▁the ▁same ▁angle ▁about ▁any ▁axis . ▁ ▁The ▁conjug ate ▁closure ▁of ▁a ▁singleton ▁containing ▁a ▁rotation ▁in ▁ 3 D ▁is ▁E +( 3 ). ▁ ▁In ▁ 2 D ▁it ▁is ▁different ▁in ▁the ▁case ▁of ▁a ▁k - fold ▁rotation : ▁the ▁conjug ate ▁closure ▁contains ▁k ▁rot ations ▁( including ▁the ▁identity ) ▁combined ▁with ▁all ▁transl ations . ▁ ▁E ( 2 ) ▁has ▁quot ient ▁group ▁O ( 2 ) ▁/ ▁C k ▁and ▁E +( 2 ) ▁has ▁quot ient ▁group ▁SO ( 2 ) ▁/ ▁C k ▁. ▁For ▁k ▁= ▁ 2 ▁this ▁was ▁already ▁covered ▁above . ▁ ▁Ref lection ▁The ▁conjug ates ▁of ▁a ▁reflection ▁are ▁ref lections ▁with ▁a ▁translated , ▁ ▁rot ated , ▁and ▁reflected ▁mirror ▁plane . ▁The ▁conjug ate ▁closure ▁of ▁a ▁singleton ▁containing ▁a ▁reflection ▁is ▁the