Jerry wrote:

But can we generate C with only one generator? The
answer is no. (Proof: Order(i)=1, Order(t)=4, Order(tt)=2, Order(tf)=3,
and Order(ttf)=2. All the orders are less than 24. Note that it suffices
to calculate the order for one representative of each conjugacy class.)

Another way to see that C cannot be generated by one generator
is to note that C is not abelian.

Singmaster mentions in his Notes that the cube group G itself
can also be generated by two elements.
--
Michiel Boland <boland@sci.kun.nl>
University of Nijmegen
The Netherlands