+--+--+--+--+
     |1 |2 |3 |4 |       
  +--+--+--+--+--+--+
  |5 |           |6 |    
  +--+--+--+--+--+--+
     |7 |8 |9 |10|
     +--+--+--+--+

1. A sat opposite G who was sitting next to C.
2. C sat opposite F.
3. D sat between F and H, but not in seat 8 or 9.
4. E sat between G and A and opposite J.
5. I sat between B and C but not in seat 9.
6. The host sat in seat 5.

DECLARE FUNCTION opp! (x$, y$)
DECLARE SUB permute (a$)
CLS
DIM SHARED a$
a$ = "abcdefghij": h$ = a$
label$ = "12346x9875"
DO
  s$ = a$ + a$
  IF opp("a", "g") THEN
    IF INSTR(s$, "gc") OR INSTR(s$, "cg") THEN
     IF opp("c", "f") THEN
      IF INSTR(s$, "fdh") OR INSTR(s$, "hdf") THEN
       IF INSTR(a$, "d") <> 8 AND INSTR(a$, "d") <> 7 THEN
        IF opp("e", "j") THEN
         IF INSTR(s$, "gea") OR INSTR(s$, "aeg") THEN
           IF INSTR(s$, "bic") OR INSTR(s$, "cib") THEN
            IF INSTR(a$, "i") <> 9 THEN
              PRINT a$
            END IF
           END IF
         END IF
        END IF
       END IF
      END IF
     END IF
    END IF
  END IF
  permute a$
LOOP UNTIL a$ = h$

FUNCTION opp (x$, y$)
  psnx = INSTR(a$, x$)
  psny = INSTR(a$, y$)
  o = 0
  SELECT CASE psnx
    CASE 5, 6
     IF psnx + psny = 15 THEN o = 1
    CASE ELSE
     IF psny = 10 - psnx THEN o = 1
  END SELECT
  opp = o
END FUNCTION

finds

hdfaegcibj

as the sequence of participants going clockwise, starting at position 1, so the position in this string does not correspond to the numbering in the original diagram.

The host is J, as the last position clockwise from 1 is the one numbered 5 in the original diagram.