r ism ▁has ▁this ▁extended ▁symmetry , ▁. ▁ ▁The ▁ch iral ▁extended ▁group ▁is ▁+ , ▁order ▁ 6 4 . ▁ ▁Ext ended ▁ch iral ▁subgroup ▁is ▁[[ 4 , 2 , 4 ]+ ], ▁order ▁ 6 4 , ▁with ▁rot ore f lection ▁gener ators . ▁It ▁is ▁is omorphic ▁to ▁the ▁abstract ▁group ▁( 4 , 4 | 2 , 4 ). ▁ ▁Other ▁extended ▁[[ 4 ], 2 , 4 ], ▁[ 4 , 2 ,[ 4 ]], ▁order ▁ 1 2 8 , ▁and ▁are ▁is omorphic ▁to ▁[ 8 , 2 , 4 ] ▁and ▁[ 4 , 2 , 8 ]. ▁The ▁ 4 – 8 ▁du op r ism ▁has ▁this ▁symmetry , ▁as ▁ ▁or ▁. ▁ ▁And ▁[[ 4 ], 2 ,[ 4 ]], ▁order ▁ 2 5 6 , ▁and ▁is ▁is omorphic ▁to ▁[ 8 , 2 , 8 ]. ▁ ▁And ▁[[ [ 4 ], 2 ,[ 4 ]] ], ▁order ▁ 5 1 2 , ▁is omorphic ▁to ▁. ▁The ▁ 8 – 8 ▁du op r ism ▁has ▁this ▁symmetry , ▁as ▁ ▁or ▁. ▁ ▁I onic ▁dimin ished ▁sub groups ▁are ▁[ 4 + , 2 , 4 ], ▁[ 4 , 2 , 4 + ], ▁order ▁ 3 2 . ▁ ▁The ▁double ▁dimin ished ▁subgroup ▁is ▁[ 4 + , 2 , 4 + ], ▁order ▁ 1 6 . ▁ ▁Ext ended ▁as ▁[[ 4 + , 2 , 4