Geodesic
Trivectors in a Space of Seven Dimensions
Canadian mathematical bulletin, Tome 20 (1977) no. 3, p. 401

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In their paper [1], Buseman and Glassco state that N(C, 7, 3) = 5 has been claimed but questioned. Schouten in [2] provides a classification of the orbits of ∧3 V under the action of the group of automorphisms of ∧3 V induced by automorphisms of V when dim V = 7. 
Westwick, R. Trivectors in a Space of Seven Dimensions. Canadian mathematical bulletin, Tome 20 (1977) no. 3, p. 401. doi: 10.4153/CMB-1977-061-5
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