▁= ▁[ 5 , 3 ]+ , ▁( ▁= ▁), ▁order ▁ 6 0 , ▁( D u ▁Val ▁# 3 1 ' ▁( I / C 1 ; I / C 1 ), ▁Con way ▁+ 1 / 6 0 [ I x I ]). ▁This ▁is ▁the ▁ 3 D ▁ch iral ▁ icos ah ed ral ▁group , ▁[ 5 , 3 ]+ . ▁ ▁A ▁sn ub ▁d ode c ah ed ral ▁py ram id ▁can ▁have ▁this ▁symmetry , ▁with ▁Sch lä fli ▁symbol : ▁(   ) ▁ ∨ ▁sr { 5 , 3 }. ▁ ▁Du op r ism atic ▁symmetry ▁ ▁Du op r ism atic ▁groups ▁– ▁[ p , 2 , q ], ▁( ), ▁order ▁ 4 p q , ▁exist ▁for ▁all ▁ 2   ≤   p , q   <   ∞ . ▁There ▁are ▁p + q ▁mirror s ▁in ▁this ▁symmetry , ▁which ▁are ▁trivial ly ▁decom posed ▁into ▁two ▁orthogonal ▁sets ▁of ▁p ▁and ▁q ▁mirror s ▁of ▁di hed ral ▁symmetry : ▁[ p ] ▁and ▁[ q ]. ▁ ▁The ▁ch iral ▁subgroup ▁is ▁[ p , 2 , p ]+ , (), ▁order ▁ 2 p q . ▁It ▁can ▁be ▁double d ▁as ▁[[ 2 p , 2 , 2 p ]+ ]. ▁ ▁If ▁p ▁and ▁q ▁are ▁equal , ▁[ p , 2 , p ], ▁( ), ▁the ▁symmetry ▁can ▁be ▁double d ▁as ▁, ▁( ). ▁