0 [ I × ]). ▁It ▁is ▁is omorphic ▁to ▁the ▁abstract ▁altern ating ▁group , ▁A 5 . ▁ ▁The ▁extended ▁ch iral ▁pent ach or ic ▁group ▁is ▁[[ 3 , 3 , 3 ]+ ], ▁order ▁ 1 2 0 , ▁( D u ▁Val ▁# 5 1 " ▁( I † / C 1 ; I / C 1 ) – † *, ▁Con way ▁+ 1 / 6 0 [ I x I ]. 2 3 ). ▁C ox eter ▁rel ates ▁this ▁group ▁to ▁the ▁abstract ▁group ▁( 4 , 6 | 2 , 3 ). ▁It ▁is ▁also ▁is omorphic ▁to ▁the ▁abstract ▁symmetric ▁group , ▁S 5 . ▁ ▁H ex ade c ach or ic ▁symmetry ▁▁▁ ▁H ex ade c ach or ic ▁group ▁– ▁B 4 , ▁[ 4 , 3 , 3 ], ▁( ), ▁order ▁ 3 8 4 , ▁( D u ▁Val ▁# 4 7 ▁( O / V ; O / V )* , ▁Con way ▁ ± 1 / 6 [ O × O ]. 2 ), ▁named ▁for ▁the ▁ 1 6 - cell ▁( hex ade c ach or on ), ▁. ▁There ▁are ▁ 1 6 ▁mirror ▁hyper plan es ▁in ▁this ▁group , ▁which ▁can ▁be ▁identified ▁in ▁ 2 ▁orthogonal ▁sets : ▁ 1 2 ▁from ▁a ▁[ 3 1 , 1 , 1 ] ▁subgroup , ▁and ▁ 4 ▁from ▁a ▁[ 2 , 2 , 2 ] ▁subgroup .