Does there exist a 1067-dimensional Lie group (say, G) with SU(12) as a subgroup, such that the quotient group $\frac{G}{SU(12)}$ happens to be of dimension 924 (= $^{12}C_6$) ?
As we know, with SU(8) there is the $E_{7(7)}$ group with a coset $\frac{E_{7(7)}}{SU(8)}$ of dimension 70 (= $^8C_4$). Similarly, with SU(4), we have $\frac{SO(1,6)}{SU(4)}$ of dimension 20 (= $^4C_2$) and so on.
So, what I am looking for is some kind of generalization for SU(N) with N=4k ( k=1, 2, 3...).