Semisymmetric cubic graphs of twice odd order

Connected cubic graphs ΓΓ of twice odd order which admit an automorphism group acting semisymmetrically are investigated. The structure of the automorphism group of ΓΓ modulo a subgroup which acts semiregularly on ΓΓ is determined. This identification is achieved by using the fundamental theorem of Goldschmidt [D.M. Goldschmidt, Automorphisms of trivalent graphs, Ann. of Math. (2) 111 (2) (1980) 377–406] and some small parts of the proof of the classification of the finite simple groups.