The quaternion group is one of the two non-Abelian groups of the five total finite groups of order 8. It is formed by the quaternions , , , and , denoted or .

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The multiplication table for is illustrated above, where rows and columns are given in the order , , , , 1, , , , as in the table above.

The cycle graph of the quaternion group is illustrated above.

The quaternion group has conjugacy classes , , , , and . Its subgroups are , , , , , and , all of which are normal subgroups.