Classification of Hadamard matrices of order 44 with automorphisms of order 7

The Hadamard matrices of order 44 possessing automorphisms of order 7 are classified. The number of their equivalence classes is 384. The order of their full automorphism group is calculated. These Hadamard matrices yield 1683 nonisomorphic 3-(44,22,10) designs, 57932 nonisomorphic 2-(43,21,10) designs, and two inequivalent extremal binary self-dual doubly even codes of length 88 (one of them being new).