Step: 183

All mirror planes are inverse to itself: ( s ) ^-1 = s , because it applies always ss = E. the same applies to E, because of EE = E. the inverse (C ₃ ² ) ^{to -1} to C ₃ ² is C ₃ , there C ₃ ² C ₃ = E (generally applies (C _n ^m ) ^-1 = C _n ^Nm ).
We saw thus that the four group axioms are fulfilled; the six symmetry operations form a group of the order 6. The group is not commutative, because it applies e.g..

s _v ⁽¹⁾ C ₃ is unequally C ₃ s _v ⁽¹⁾ ,
because s _v ⁽¹⁾ C ₃ = s _v ⁽²⁾ and C ₃ s _v ⁽¹⁾ = s _v ⁽³⁾

Go now to training step 117 !