SQL query with rand select

Please, can I have some help with a MySQL query? I have a list of 18 entries in a table with each a corresponding group, there are 6x groups. My requirement is to randomly pick 6 entries from that table and only take one entry per group.

Here is my current query, but the problem with this is it is not taking only 1 entry per group, I am getting 2 results for group 3 returned for example.

SELECT mask,group_id, icon, infoText FROM table_name ORDER BY RAND() LIMIT 6;

Current output when running above query, as you can see 2x results for group 1 are returned.

mask	group_id	icon	infoText
2	1	dot	blaba
7	1	dot	blaba2
6	2	dot	blaba3
4	3	dot	blaba4
9	4	dot	blaba5
10	5	dot	blaba6

When I want it to something like this instead, when a random mask is selected but only 1 entry from that group

mask	group_id	icon	infoText
1	1	dot	blaba
2	2	dot	blaba2
6	3	dot	blaba3
3	4	dot	blaba4
5	5	dot	blaba5
12	6	dot	blaba6