▁are ▁given ▁in ▁C ox eter ▁notation , ▁which ▁are ▁based ▁on ▁C ox eter ▁groups , ▁with ▁mark ups ▁for ▁extended ▁groups ▁and ▁sub groups . ▁C ox eter ▁notation ▁has ▁a ▁direct ▁correspond ence ▁the ▁C ox eter ▁diagram ▁like ▁[ 3 , 3 , 3 ], ▁[ 4 , 3 , 3 ], ▁[ 3 1 , 1 , 1 ], ▁[ 3 , 4 , 3 ], ▁[ 5 , 3 , 3 ], ▁and ▁[ p , 2 , q ]. ▁These ▁groups ▁bound ▁the ▁ 3 - s phere ▁into ▁identical ▁hyp ers pher ical ▁t etra hed ral ▁domains . ▁The ▁number ▁of ▁domains ▁is ▁the ▁order ▁of ▁the ▁group . ▁The ▁number ▁of ▁mirror s ▁for ▁an ▁ir reducible ▁group ▁is ▁n h / 2 , ▁where ▁h ▁is ▁the ▁C ox eter ▁group ' s ▁C ox eter ▁number , ▁n ▁is ▁the ▁dimension ▁( 4 ). ▁ ▁For ▁cross - refer encing , ▁also ▁given ▁here ▁are ▁qu atern ion ▁based ▁not ations ▁by ▁Patrick ▁du ▁Val ▁( 1 9 6 4 ) ▁and ▁John ▁Con way ▁( 2 0 0 3 ). ▁Con way ' s ▁notation ▁allows ▁the ▁order ▁of ▁the ▁group ▁to ▁be ▁computed ▁as ▁a ▁product ▁of ▁elements ▁with ▁ch iral ▁poly hed ral ▁group ▁orders : ▁( T = 1 2 , ▁O = 2 4 , ▁I = 6 0 ). ▁In ▁Con way ' s ▁notation , ▁a ▁( ± ) ▁prefix ▁implies ▁central ▁in version