Hard battles followed. At last, west of St.Lô, the Americans broke through the German lines in August 1944. An armoured counterattack ordered by Hitler was broken up by British rocket-firing Typhoon fighter-bombers and American tanks. As the Germans retreated west, they were trapped in the “Falaise Pocket” between the British and Canadians to the north and the onrushing Americans to the south. Although the Canadians were slow to close the gap at the east end of the pocket, some 50,000 German soldiers were killed and 200,000 taken prisoner. The German army lost most of its armour and vehicles in France.
3rd Canadian Infantry Division and 2nd Canadian Armoured Brigade formed the Canadian assault force on D-Day, while 1st Canadian Parachute Battalion jumped as part of the great airborne force protecting the flanks of the landings. Canadian destroyers, corvettes, minesweepers, landing ships and landing craft supported the landings, as did the many RCAF squadrons overhead. At first, the Canadians came ashore under British command, but with the arrival of 2nd Canadian Infantry Division and 4th Canadian Armoured Division in July, all came under the command of their own leaders in the First Canadian Army.
Reference: https://www.warmuseum.ca/cwm/exhibitions/newspapers/operations/ddaynormandy_e.html