Geass Order

The Geass Order, also translated as the Geass Cult or Geass Directorate, is a secretive group that studies and produces Geass users. Its last known location is a large underground city, also called the Geass Order, which is within the Chinese Federation territory. Before the assault instigated by the Black Knights, it was under the control of V.V., and so was presumably under Britannian influence.

Description
Origins of The Geass Directorate are unknown but since it's former leader was C.C. in Code Geass Akito the Exiled, it is implied that it's former headquarters was in E.U. The Geass Directorate plays a major role in Code Geass: Oz The Reflection due to it's known members.

Though its location usually changes with each leader, it is currently located in an enormous underground cavern beneath an unspecified desert somewhere in the Chinese Federation. The Geass Order seems to be, for the most part, a fairly modern medium-sized city, complete with streetlights, roadways, ornate buildings, and even an underground train system. The location of the directorate may be the Tengger Desert in our Inner Mongolia, which resembles the desert as seen in Code Geass. Another location of the Geass Directorate was said to be in E.U. territory despite C.C.'s departure during the member's escape from the Black Knight's assault.

Most of the city is lit with an eerie purple glow that emanates from a Thought Elevator at the head of the city.

Assault on the Geass Order
After the death of Shirley Fenette, an embittered Zero decides to launch an attack on the order as to destroy the existence of Geass as a whole. Crashing through the shallow roof of the cavern, the Black Knights, on Zero's orders, indiscriminately slaughter the city's unarmed inhabitants. While V.V. attempts to stave off the attack with the Siegfried, a group of hooded order members attempt to escape the city with essential Geass research in a bullet train. C.C., however, intervenes and destroys the train before it can exit the compound.

Lelouch, pursuing the wounded (though immortal) V.V. to the Thought Elevator, in which the Emperor can be seen standing. After the Emperor and V.V. exchange a few words, Lelouch finds himself suddenly transported out of the city and onto the Sword of Akasha. Some of the order agents can be seen reporting to Charles zi Britannia after V.V's death implying he's their new leader.

Unnamed Children
When Zero led the Black Knights to destroy the Geass Order, some unnamed children were present who referred to Rolo as "Big Brother Rolo". One of them displayed a Geass power, able to control another person's body. All of the children were killed by Rolo.

Nightmare of Nunnally
The Geass Order in Nightmare of Nunnally is similar but slightly different. They go by the name Eden Vital Religious Order whose goal is dedicated to unleashing the Heaven's Door (which is in fact Ragnarok Connection) to bring humanity into one being. Their organization is in fact highly structured where they have different divisions such as Heretical Trial Division with Cardinals running them and Rolo Vi Britannia is one example. Like in the anime, it is implied that they research Code R, duplicating C.C.'s powers furthering Geass abilities. They also deal with the use of witchcraft as they once tend to burn Nunnally for her crimes by Rolo's orders of the Heretical Trial Division only to be stopped by the likes of Lelouch, Suzaku, and Euphemia. It implies they deal with religious like use techniques to demonstrate their organization's philosophy. Their role was active when Charles Zi Britannia declares the Holy Nation of Eden Vital to replace Britannia and as a result, all Cardinals were to replace Viceroys and have them under arrest to ensure Heaven's Door's fruition. They later battled against  the combined Britannia and Black Knights at Kaminejima island only to be defeated when Charles and Marianne were rejected by Nunnally's choice (similar to Lelouch in the anime) to follow the future instead of living in Heaven's Door which was the past. Afterwards, the nation no longer existed.