R - Calculate difference for each first value in group

Question

I have the following dataframe in R:

#dataframe
data <- cbind.data.frame(Line = 1:6,
                         Group = c(rep("A", 3), rep("B", 3)),
                         Value = c(1, 2, 4, 1, 2, 3))

So the table looks like this:

Line Group Value
1     A     1
2     A     2
3     A     4
4     B     1
5     B     2
6     B     3

Now I want to calculate for each group, the difference to the first value in the group (which is a control value), so:

Line 1 - Line 1
Line 1 - Line 2
Line 1 - Line 3
Line 4 - Line 4
Line 4 - Line 5
Line 4 - Line 6

I think this should work in a loop but I wanted to ask if there is maybe an easier way to that calculation? Thank you very much!