PostgreSQL. Select a column that correlates with value in the aggregate function

Here is the 'items' table, containing more than 10 rows:

+-----+-----------+-----------+----------+
| id  | item_name | category  | quantity |
+=====+===========+===========+==========+
| 3   | item33    | category1 | 5        |
+-----+-----------+-----------+----------+
| 2   | item52    | category5 | 1        |
+-----+-----------+-----------+----------+
| 1   | item46    | category1 | 3        |
+-----+-----------+-----------+----------+
| 4   | item11    | category3 | 2        |
+-----+-----------+-----------+----------+
| ... | ...       | ...       | ...      |
+-----+-----------+-----------+----------+

Values in the 'items' column are unique, the ones in the 'category' columnt - aren't unique.

The task is:

1. Remove duplicates of categories: if a category contains more than 1 item, take the row with minimal 'id'.
2. Order results by the 'quantity' (ASC).
3. Take 10 rows: top 5 and random 5 from the rest result data output.

So, the ordering table (after #2 sub-task) should look like that:

+-----+-----------+-----------+----------+
| id  | item_name | category  | quantity |
+=====+===========+===========+==========+
| 2   | item52    | category5 | 1        |
+-----+-----------+-----------+----------+
| 4   | item11    | category3 | 2        |
+-----+-----------+-----------+----------+
| 1   | item46    | category1 | 3        |
+-----+-----------+-----------+----------+
| ... | ...       | ...       | ...      |
+-----+-----------+-----------+----------+

I know how to exclude duplicates for categories:

SELECT min(id) as id, category
FROM items
GROUP BY category

But I don't know how to order it by the quantity. If I try to add 'quantity' to the 'select' line and then make 'ORDER BY quantity', I get the error: "column "quantity" must appear in the GROUP BY clause or be used in an aggregate function".

If there is a way to add this 'quantity' column to the data output (the value in this column should correlate with the resulting 'id' value (i.e. "min(id)"))? And then do ordering and picking rows...