Group by multiple variables

Hello,

I would like to create a group variable that is based on the values of three variables:

For example,

dat <- data.frame(A=c(1,1,1,1,1,2,2,2,2,2),

? ? ? ? ? ? ? ? ? ?B=c(1,1,1,5,5,5,9,9,9,9),
? ? ? ? ? ? ? ? ? ?C=c(1,1,1,1,1,2,2,7,7,7))

dat

? A B C
?1 ?1 1 1
?2 ?1 1 1
?3 ?1 1 1
?4 ?1 5 1
?5 ?1 5 1
?6 ?2 5 2
?7 ?2 9 2
?8 ?2 9 7
?9 ?2 9 7
10 2 9 7

All rows that that are equal, belong to the same group, that is I would like to create a group variable like this:

? A B C group
?1 ?1 1 1 ? 1
?2 ?1 1 1 ? 1
?3 ?1 1 1 ? 1
?4 ?1 5 1 ? 2
?5 ?1 5 1 ? 2
?6 ?2 5 2 ? 3
?7 ?2 9 2 ? 4
?8 ?2 9 7 ? 5
?9 ?2 9 7 ? 5
10 2 9 7 ? 5

Is there an easy way to do this? Right now I use a bunch of loops and that seems rather cumbersome. In general, my data set is rather large and the number of categories per variable is not always fixed.

Franco