18.86 (2014)

Yes, it is linear, since it embeds in the holomorph $\text{Hol } F_2$ of the free group $F_2$, which can be seen by adding a new generator $c = [a, b]$, so that $G = \langle a, b, c \mid a^b = ac, c^b = c \rangle$, while $\text{Hol } F_2$ was shown to be linear in Corollary 3 in (V. G. Bardakov, O. V. Bryukhanov, Vestnik Novosibirsk Univ. Ser. Mat. Mekh. Inf., 7, no. 3 (2007), 45–58 (Russian)) (O. V. Bryukhanov, Letter of 27 January 2014). Another proof can be found in (V. A. Roman'kov, J. Siberian Federal Univ. Math. Phys., 6, no. 4 (2013), 516–520).