Hello Victor,
AXC is for Adrien Chobillon, a famous parisian manufacturer (from 1912 until 1979 when sold to Arthus Bertrand).
IIIrd Republic (1870-1940) Legion of Honour were made between 1870 and 1951, I'm not sure it's possible to date them exactly.
1870 on the averse
IVth Republic (1946-1958) LoH were made between 1951 and 1962
star on the averse
Vth Republic (since 1958) are made since 1962.
star on the averse
"29 Floréal an X" on the observe (date of the creation of the LoH in the republican calendar)
As you can see, to call them IIIrd, IVth ot Vth RP is not exact as the dates of manufacturing do not coincide exactly with the dates of each Republic, it's more a collector name.
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Cordialement,
François-Xavier