Confused about F-contrast and F-test in statistical software

Let's say I have an estimated parameter $\beta$ from a general linear model for 2 subject groups (with the $\beta$s corresponding to their means).

If I wanted to do a two sample t test in a particular way I can define a contrast with which I can formulate the t score as:

$$T = \frac{contrast\;of\;estimated\;parameters}{sqrt(variance\;estimate)} = \frac{c'\beta}{sqrt(variance\;estimate)}$$

where c' is a row vector (for example $[-1\;1]$), and $\beta$ is a column vector $[ \beta_1\;\beta_2]^T$. So for the t score one gets a scalar number.

My question is what happens when I have 3 groups and I would like to test multiple contrasts. I gather I should use an ANOVA which uses the F test rather than the t-test. But I don't quite understand how a statistical software would visualize an F score if one is testing for the overall differences between the groups using an F contrast. For example:

$$c' = \begin{bmatrix} 1 &-1 &0 \\ 0 & 1 & -1 \end{bmatrix}$$

Is there a way to formulate the F score for 3 groups like in the case of the t score for 2 groups.(the estimated parameters being $[ \beta_1\;\beta_2\;\beta_3]^T$ ) I would expect the F score to be a vector in this case. Am I wrong in assuming this?

Many thanks