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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[1] 1. Buseman, H. and Glassco, D. E., Irreducible Sums of Simple Multivectors, Pac. J. of Math., vol. 49, no. 1, 1973, (13-32). Google Scholar

[2] 2. Schouten, J. A., Klassifizierung der altemierenden Gröszen Dritten Grades in 7 Dimensionen. Cir. Mat. di Palermo, vol. 55, 1931, (137-156). Google Scholar

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