▁group . ▁It ▁is ▁is omorphic ▁to ▁the ▁abstract ▁symmetric ▁group , ▁S 5 . ▁The ▁extended ▁pent ach or ic ▁group , ▁Aut ( A 4 ), ▁, ▁( The ▁dou bling ▁can ▁be ▁hint ed ▁by ▁a ▁fol ded ▁diagram , ▁), ▁order ▁ 2 4 0 , ▁( D u ▁Val ▁# 5 1 ▁( I † */ C 2 ; I / C 2 ) † *, ▁Con way ▁ ± 1 / 6 0 [ I × ]. 2 ). ▁It ▁is ▁is omorphic ▁to ▁the ▁direct ▁product ▁of ▁abstract ▁groups : ▁S 5 × C 2 . ▁ ▁The ▁ch iral ▁extended ▁pent ach or ic ▁group ▁is ▁+ , ▁( ), ▁order ▁ 1 2 0 , ▁( D u ▁Val ▁# 3 2 ▁( I † / C 2 ; I / C 2 ) † , ▁Con way ▁ ± 1 / 6 0 [ I x ]). ▁This ▁group ▁represents ▁the ▁construction ▁of ▁the ▁om nis n ub ▁ 5 - cell , ▁, ▁although ▁it ▁can ▁not ▁be ▁made ▁uniform . ▁It ▁is ▁is omorphic ▁to ▁the ▁direct ▁product ▁of ▁abstract ▁groups : ▁A 5 × C 2 . ▁The ▁ch iral ▁pent ach or ic ▁group ▁is ▁[ 3 , 3 , 3 ]+ , ▁( ), ▁order ▁ 6 0 , ▁( D u ▁Val ▁ ▁# 3 2 ' ▁( I † / C 1 ; I / C 1 ) † , ▁Con way ▁+ 1 / 6