His name on both the 1850 & 60 census show only his initials as that for his parents. In 1860 we find

Father G. W. Harmon, 57, Farmer & Tanner
Mother Nancy Harmon, 54
F. C. Harmon, 19 Farmer, born in Georgia

Residents of: Elberton, Elbert, Georgia, Grove Post Office
Ancestry.com

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F C. Harmon

Residence Elbert County GA;
Enlisted on 2/25/1862 as a Private.
On 2/25/1862 he mustered into "C" Co. GA 15th Infantry
He died of wounds on 5/9/1864
He was listed as:
* Wounded 5/6/1864 Wilderness, VA

Sources used by Historical Data Systems, Inc.:
- Roster of Confederate Soldiers of Georgia 1861-1865