# 1.6: Constructing higher groups from simpler groups

• • Contributed by Claire Vallance
• Professor of Physical Chemistry (Department of Chemistry) at University of Oxford

A group that contains a large number of symmetry elements may often be constructed from simpler groups. This is probably best illustrated using an example. Consider the point groups $$C_2$$ and $$C_S$$. $$C_2$$ contains the elements $$E$$ and $$C_2$$, and has order 2, while $$C_S$$ contains $$E$$ and σ and also has order $$2$$. We can use these two groups to construct the group $$C_{2v}$$ by applying the symmetry operations of $$C_2$$ and $$C_S$$ in sequence.

$\begin{array}{lllll} C_2 \: \text{operation} & E & E & C_2 & C_2 \\ C_S \: \text{operation} & E & \sigma(xz) & E & \sigma(xz) \\ \text{Result} & E & \sigma_v(xz) & C_2 & \sigma_v'(yz) \end{array} \tag{6.1}$

Notice that $$C_{2v}$$ has order $$4$$, which is the product of the orders of the two lower-order groups. $$C_{2v}$$ may be described as a direct product group of $$C_2$$ and $$C_S$$. The origin of this name should become obvious when we review the properties of matrices.