I wasn't able to get some of the other answers to work on Chrome, Firefox, IE11, and Edge with the same code.
I came up with the below and it appears to work for those browsers listed above. I also wanted to see what OS the user was on. I haven't tested this against a browser with user overridden User-Agent settings, so mileage may vary. The order of the IFs is important for this to work correctly.
let os, osStore, bStore, appVersion, browser;
// Chrome
if(navigator.vendor === "Google Inc."){
    appVersion = navigator.appVersion.split(" ");
    os = [appVersion[1],appVersion[2],appVersion[3],appVersion[4],appVersion[5]].join(" ");
    os = os.split("(")[1].split(")")[0]
    browser = appVersion[appVersion.length-2].split("/").join(" ");
    console.log("Browser:",browser,"- OS:",os);
}
// Safari
else if(navigator.vendor === "Apple Computer, Inc."){
    appVersion = navigator.appVersion.split(" ");
    os = [appVersion[1],appVersion[2],appVersion[3],appVersion[4],appVersion[5]].join(" ");
    os = os.split("(")[1].split(")")[0];
    browser = appVersion[appVersion.length-1].split("/").join(" ");
    console.log("Browser:",browser,"- OS:",os);
}
// Firefox is seems the only browser with oscpu
else if(navigator.oscpu){
    bStore = navigator.userAgent.split("; ").join("-").split(" ");
    browser = bStore[bStore.length-1].replace("/"," ");
    osStore = [bStore[1],bStore[2],bStore[3]].join(" ");
    osStore = osStore.split("-");
    osStore.pop(osStore.lastIndexOf)
    osStore = osStore.join(" ").split("(");
    os = osStore[1];
    console.log("Browser:",browser,"- OS:",os);
}
// IE is seems the only browser with cpuClass
// MSIE 11:10 Mode
else if(navigator.appName === "Microsoft Internet Explorer"){
    bStore = navigator.appVersion.split("; ");
    browser = bStore[1]+" / "+bStore[4].replace("/"," ");
    os = [bStore[2],bStore[3]].join(" ");
    console.log("Browser:",browser,"- OS:",os);
}
// MSIE 11
else if(navigator.cpuClass){
    bStore = navigator.appVersion.split("; ");
    osStore = [bStore[0],bStore[1]].join(" ");
    os = osStore.split("(")[1];
    browser = "MSIE 11 "+bStore[2].split("/").join(" ");
    console.log("Browser:",browser,"- OS:",os);
}
// Edge
else if(navigator.appVersion){
    browser = navigator.appVersion.split(" ");
    browser = browser[browser.length -1].split("/").join(" ");
    os = navigator.appVersion.split(")")[0].split("(")[1];
    console.log("Browser:",browser,"- OS:",os);
}
// Other browser
else {
    console.log(JSON.stringify(navigator));
}