Looking at info on the web, the 34th Sep. Co. was in the New York National Guard, recruited from the Geneva, NY area, and more particularly, it seems, from local fire companies. It seems to have formed in 1880. There's a good bit of information on the internet, particularly the NY DMNA site with link below.
This paragraph is from that site:
3rd Regiment Infantry
New York Volunteers
The 3rd Infantry Regiment was one of twelve New York State National Guard units federalized for service in the Spanish-American War. The 3rd Regiment was created from a mixture of twelve separate companies drawn from the ranks of 4th Brigade of the New York National Guard. The twelve companies were brought together on May 2nd, 1898 at Camp Black. Upon their arrival at the camp they were grouped into three battalions. The First Battalion consisted of 30t, 43rd, 47th, and 48th Separate Companies, which were re-designated companies L, I, K, and D respectively. The Second Battalion contained the 2nd, 8th, 34th, and 41st Separate Companies, renamed companies M, A, B, and C. Finally the Third Battalion was composed of the 1st, 25th, 29th, and 42nd Separate Companies, which became companies H, G, F, and E. The regiment drew its membership from a number of upstate communities. Companies A and H came from Rochester and companies B, C, D, and E came from Geneva, Syracuse, Oswego, and Niagara Falls respectively. Companies F, G, I, K, L, and M were recruited from Medina, Tonawanda, Olean, Hornellsville, Elmira, and Auburn respectively. The regiment was officially mustered in on May 17th, 1898 and designated the “3rd Regiment Infantry, New York State Vols.”
New York and the War with Spain: History of the Empire State Regiments. Albany: Argus Company Printers, 1903.
Internet pages with info:
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