What's this gate?

It is a NAND gate.

The truth table of NAND is shown below,
where 1 = true, and 0 = false

I don't know of any basic gate like that. You would have to build a compound gate from basic ones. I think it could be done with 6 gates. Gates #1 - #3 are NOT to produce the opposites of the inputs. Gate #4 is a 3-input AND of a, b, c. Gate #5 is a 3-input AND of -a, -b and -c. Then gate #6 is a 2-input NOR of the outputs of gates #4 and #5.

There are devices which can be configured for any logic function of the inputs, such as FPGA. Trouble is, they have to be programed, which is a lot of complexity.

There is no standard single gate with that specific function.
Nothing stops you making one from other gates.

One way would be a three-input decoder such as a 74HC193, with a diode AND gate on outputs 0 and 7 (two diodes, cathodes to the 193 outputs, anodes joined and with a pull-up resistor).

That would give the low (false) output for 000 & 111, high for any other value.

You could also use a three input OR and a three input NAND, combined using an AND gate (functioning as a negative logic OR) to give the output.

Again you could use the diode AND, but if you are using a NAND IC you can use two gates as NAND followed by NOT (all inputs joined) to give the extra AND function.

NAND gate in part of the equation