▁be ▁described ▁as ▁applying ▁the ▁is ometry ▁to ▁the ▁translation : ▁the ▁conjug ate ▁of ▁a ▁translation ▁by ▁a ▁translation ▁is ▁the ▁first ▁translation ▁the ▁conjug ate ▁of ▁a ▁translation ▁by ▁a ▁rotation ▁is ▁a ▁translation ▁by ▁a ▁rot ated ▁translation ▁vector ▁the ▁conjug ate ▁of ▁a ▁translation ▁by ▁a ▁reflection ▁is ▁a ▁translation ▁by ▁a ▁reflected ▁translation ▁vector ▁ ▁Thus ▁the ▁conjug acy ▁class ▁within ▁the ▁E uclidean ▁group ▁E ( n ) ▁of ▁a ▁translation ▁is ▁the ▁set ▁of ▁all ▁transl ations ▁by ▁the ▁same ▁distance . ▁ ▁The ▁smallest ▁subgroup ▁of ▁the ▁E uclidean ▁group ▁containing ▁all ▁transl ations ▁by ▁a ▁given ▁distance ▁is ▁the ▁set ▁of ▁all ▁transl ations . ▁Thus ▁this ▁is ▁the ▁conjug ate ▁closure ▁of ▁a ▁singleton ▁containing ▁a ▁translation . ▁ ▁Thus ▁E ( n ) ▁is ▁a ▁sem id irect ▁product ▁of ▁the ▁orthogonal ▁group ▁O ( n ) ▁and ▁the ▁subgroup ▁of ▁transl ations ▁T , ▁and ▁O ( n ) ▁is ▁is omorphic ▁with ▁the ▁quot ient ▁group ▁of ▁E ( n ) ▁by ▁T : ▁O ( n ) ▁ ▁E ( n ) ▁/ ▁T ▁ ▁Thus ▁there ▁is ▁a ▁partition ▁of ▁the ▁E uclidean ▁group ▁with ▁in ▁each ▁subset ▁one ▁is ometry ▁that ▁keeps ▁the ▁origin ▁fixed , ▁and ▁its ▁combination ▁with ▁all ▁transl ations . ▁ ▁Each ▁is ometry ▁is ▁given ▁by ▁an ▁orthogonal ▁matrix ▁A ▁in ▁O ( n ) ▁and ▁a ▁vector ▁b : ▁ ▁and ▁each ▁subset ▁in ▁the ▁quot ient ▁group ▁is ▁given ▁by