▁du op r ism atic ▁group ▁– ▁[ 2 , 2 , 2 ], ▁( ), ▁order ▁ 1 6 . ▁ ▁The ▁ch iral ▁subgroup ▁is ▁[ 2 , 2 , 2 ]+ , ▁( ), ▁order ▁ 8 . ▁ ▁Ext ended ▁, ▁( ), ▁order ▁ 3 2 . ▁The ▁ 4 - 4 ▁du op r ism ▁has ▁this ▁extended ▁symmetry , ▁. ▁ ▁The ▁ch iral ▁extended ▁group ▁is ▁+ , ▁order ▁ 1 6 . ▁ ▁Ext ended ▁ch iral ▁subgroup ▁is ▁[[ 2 , 2 , 2 ]+ ], ▁order ▁ 1 6 , ▁with ▁rot ore f lection ▁gener ators . ▁It ▁is ▁is omorphic ▁to ▁the ▁abstract ▁group ▁( 4 , 4 | 2 , 2 ). ▁ ▁Other ▁extended ▁[( 3 , 3 )[ 2 , 2 , 2 ]] =[ 4 , 3 , 3 ], ▁order ▁ 3 8 4 , ▁# H ex ade c ach or ic ▁symmetry . ▁The ▁t esser act ▁has ▁this ▁symmetry , ▁as ▁ ▁or ▁. ▁ ▁I onic ▁dimin ished ▁sub groups ▁is ▁[ 2 + , 2 , 2 ], ▁order ▁ 8 . ▁ ▁The ▁double ▁dimin ished ▁subgroup ▁is ▁[ 2 + , 2 , 2 + ], ▁order ▁ 4 . ▁ ▁Ext ended ▁as ▁[[ 2 + , 2 , 2 + ]], ▁order ▁ 8 . ▁ ▁The ▁rot ore f lection ▁sub groups ▁are ▁[ 2 + , 2 + , 2 ], ▁[ 2 ,