The definition of Chief series is given in a link.
Question:
Consider the group G= $\mathbb{Q}_8 \rtimes C_3 $ (where the action is non trivial), its chief series is $$ e \triangleleft Z_2 \triangleleft \mathbb{Q}_8 \triangleleft G.$$
Is the following a Chief series for the group $H=(\mathbb{Q}_8 \rtimes C_3 ) \times A_5$?
$$ e \triangleleft (e \times A_5) \triangleleft (Z_2 \times A_5) \triangleleft (\mathbb{Q}_8 \times A_5) \triangleleft H.$$
If this is a chief series for $H$ then my question is why? If this is not then what is a correct series?
Thanks!
Edit: As @Derek Holt suggested, I have made changes in the question. Thank you!