6th Armored Infantry was organic to 1st Armored Division. Company C was an element of the regiment's 1st Battalion - one of the battalion's three rifle companies. The regiment was constituted in January 1812 and organized two months later but I won't go quite that far back. The regiment was reorganized as armored infantry and assigned to the 1st Armored Division in July 1940. Overseas 31 May 1942, arriving in Ireland 11 June, where it re-equipped and trained prior to landing near Oran, Algeria as one of the assault elements on 8 November. Moved to Italy October 1943 where it served throughout the remainder of the war.
1st Battalion was reorganized and redesignated 6th Armored Infantry Battalion effective from 20 July 1944, just a few days subsequent to your cousin's death. The regiment was awarded the Presidential Unit Citation for action at Oran, Algeria and a second award for action at Mount Porchia, Italy in January 1944. Campaign credits: Algeria-French Morocco (Invasion), Algeria-French Morocco (Campaign), Tunisia, Naples-Foggia, Anzio, Rome-Arno, North Apennines and Po Valley. Frazier died within the time frame of the Rome-Arno Campaign. How any of the rest of the above applies depends on his dates of service with the regiment and other factors.
The division veterans association may be able to assist in locating individuals who served with him or in securing other inforamtion:
1st Armored Division Association Post Office Box 2088 Elizabethtown, Kentucky 42702-2088
His deceased file may contain the location and, perhaps, circumstances of his death. You'll find instructions at Message #2112.