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We calculate the character table of a sharply $5$-transitive subgroup of $\alter(12)$‎, ‎and of a sharply $4$-transitive subgroup of $\alter(11)$‎. ‎Our presentation of these calculations is new because we make no reference to the sporadic simple Mathieu groups‎, ‎and instead deduce the desired character tables using only the existence of the stated multiply transitive permutation representations‎.