Sam Loyd's Keys Puzzle

Bluebeard explains that his bunch of keys was strung upon an endless key ring and divided into three groups so that the first group multiplied by the second equaled the third!

That was the secret by which he knew whether the keys had been tampered with and forbidden chambers had been entered.

You see that 6910 multiplied by 7 does not amount to 83452, so the keys were not replaced properly in their groups.

Can our clever puzzlists show how the keys must have been arranged in three groups so that the first group multiplied by the second makes the third?