How to implement C++ code in order to be similar to Lua table?

I am here for help because I'm trying to setup array table that is similar to my lua table like this:

game = {
    Name = "game",
    Characters = {
        Name = "Characters"
    },
}

Still, It gives me nill from game.Characters.Name, and game.Name changes its variable into Characters by "lua_pushstring(lua_state, "Characters")"

Here's my unfinished C++ code where I want it to be similar to the table in Lua code

luaL_newmetatable(lua_state, "Global");
int mainGameindex = lua_gettop(lua_state);
lua_pushvalue(lua_state, mainGameindex);
std::cout << "INDEX " << mainGameindex << std::endl; //This is a debug message
lua_setglobal(lua_state, "game"); //-> game = {}, it's a table

lua_pushstring(lua_state, "game");
lua_setfield(lua_state, -2, "Name"); // -> game.Name = "game"

lua_newtable(lua_state);
lua_setfield(lua_state, -2, "Characters"); // --> game.Characters = {}, it's a table again

lua_pushstring(lua_state, "Name");
lua_pushstring(lua_state, "Characters");
lua_settable(lua_state, -3); 

My thought was "game.Characters.Name = "Characters"". Without it, it would also work.

So, Can anyone point out what am I doing wrong, or just replace my C++ code with your fixed code.

EDIT: I re-explain because after I tried to start debug, no error, but still a problem because game.Name changes its variable from "game" to "character".

lua_settable(lua_state, -3);

Here you assume that you have the table at index -3, but you don't.
Previous lua_setfield has popped this table from the stack.
You need to explicitly push another reference to game.Characters before inserting a field into it.