The Oh character table has ten symmetry classes. I am trying to identify them, but my stumbling block is in identifying the order of each class.
I will use this diagram as a reference, from Sydney Kettle's Symmetry and Structure: Readable Group Theory for Chemists pg 191 (Wiley, 3rd edition).
The classes (with their respective orders) for the Oh table are:
$$\begin{array}{cccccccccc}E & 8C_3 & 6C'_2 & 6C_4 & 3C_2 & i & 6S_4 & 8S_6 & 3\sigma_\mathrm{h} & 6\sigma_\mathrm{d}\end{array}$$
$E$ and $i$ are obvious.
$6C'_2$ and $3C_2$ are shown in the diagram.
$8C_3$ is shown in the diagram above; however, it only displays 4 axes. The character table indicates there are 8 equivalent rotations for this symmetry operation. Where do the other 4 axes for the $C_3$ class come from?
$6C_4$ is the same issue. It's only shown as 3 axes. Where are the other 3 $C_4$ axes from?
The improper rotations are given in another diagram, this time on page 194 from the same book.
The mirror and dihedral planes, $\sigma_\mathrm{h}$ and $\sigma_\mathrm{d}$, are both shown to match the order in the character table.
Similar to above for $C_3$ and $C_4$, the $S_4$ and $S_6$ rotations are only half the order supplied in the character table.
In summary, from 10 total classes in the octahedral table, I can identify all the corresponding axes/planes for 6 of those 10 classes. For the remaining 4 classes, $C_3$, $C_4$, $S_4$, and $S_6$, I can only identify half the axes. At first I thought the character table was counting both positive and negative rotations, but then the order for the $C'_2$ and $C_2$ rotations should be doubled if that were the case ($12C'_2$ and $6C_2$).
Can anyone please advise on exactly what the $C_3$, $C_4$, $S_4$, and $S_6$ operations correspond to in an octahedral complex?