| Summary: | <p>We show that there are exactly 4285 symmetric (45,12,3) designs that admit nontrivial automorphisms.<br /> Among them there are 1161 self-dual designs and 1562 pairs of mutually dual designs.<br /> We describe the full automorphism groups of these designs and analyze their ternary codes.<br /> R. Mathon and E. Spence have constructed 1136 symmetric (45,12,3) designs with trivial automorphism group,<br /> which means that there are at least 5421 symmetric (45,12,3) designs.<br /> Further, we discuss trigeodetic graphs obtained from the symmetric $(45,12,3)$ designs.<br /> We prove that $k$-geodetic graphs constructed from mutually non-isomorphic designs<br /> are mutually non-isomorphic, hence there are at least 5421 mutually non-isomorphic trigeodetic graphs<br /> obtained from symmetric $(45,12,3)$ designs.</p>
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