Generating sets for subgroups of symmetric group:S4

This article provides summary information on various choices of generating set for subgroups of symmetric group:S4. It builds on basic information available at element structure of symmetric group:S4 and subgroup structure of symmetric group:S4.

Probability of generation
The rule is as follows. Given $$k$$ (not necessarily distinct) elements picked uniformly at random and independently of each other from a finite group $$G$$, the probability that they all live in a fixed subgroup of index $$d$$ is $$1/d^k$$.

Using this and a form of Mobius inversion on the subgroup lattice, it is possible to compute the probability that they generate a fixed subgroup of index $$d$$ (we basically need to subtract off probabilities for smaller subgroups).

Generated by one element
Here, a single element is picked uniformly at random from the group.

Generated by two independent possibly equal elements
Here, two elements are picked uniformly at random from the group, independent of each other. They could be equal.

Small generating sets for subgroups
We know that the subgroup rank of the symmetric group of degree four is 2, i.e., all its subgroups are 2-generated groups. On the other hand, we also know that it is possible to come up with small generating sets that are not of minimum size. Some information on this is included below.

For brevity, we do not include all subsets of interest but describe our subsets by means of generic descriptors.

(More rows need to be filled in)