The Super Galaxy Dai-Gurren (超銀河ダイグレン, Choginga Daiguren?), christened by Simon, is colossal battleship-type Gunmen akin to the Arc-Gurren which first appeared in the prologue of the first episode before being formally introduced near the end of the third part of the series.
Originally, it was called the Cathedral Terra and it was Lordgenome's flag ship against the Anti-Spirals in the past until Lordgenome was turned to the will of the enemy and was made King of Earth. It then assumed the form of the moon, acting as a sentry to report any Spiral activity on Earth.
This was until the climatic battle during the Anti Spirals' attack on the Arc-Gurren reveals its true identity and attaks the Spirals. After Simon defeats and reclaims it for the Spirals and Team Dai-Gurren, it becomes their flagship for the final part of the series on their crusade to defeat the Anti-Spirals and save Nia, and from that point foward. Because of its size, the people on board use transportation technology to move around the ship. Its combat abilities are unparalleled by any other battleship-type mecha in the series, sporting an enormous array of missiles, beam weapons and shielding capabilities. It also has the capability to transverse to other dimensions and universes and has the manufacturing capabilities to create hundreds of Arc-Gurren Lagann -sized mecha in a small period of time.
During the final battle with the Anti-Spirals, when dealing with larger Ashtanga, Simon channeled enough massive Spiral Energy, with aid from the spiral energy of those who came before him and hi-jacked it to transform the battleship into a humanoid shape and wiped out the opposition. This Gunmen was named by him, the Trascendant Super Spiral Dreadnought Gunmen Super Galaxy Gurren Lagann (超銀河グレンラガン, Choginga Guren-Ragan?).
The Super Galaxy Gurren Lagann is under half the size of Earth. It is piloted by a cascading control interface. The system goes as such: Simon pilots the Gurren Lagann, which is in turn pilots the Arc-Gurren Lagann, mimicking his moves, which controls the Super Galaxy Gurren Lagann. Because it is so massive to be controlled by a single Gunman, it contains the bridge originally seen on the Super Galaxy Dai-Gurren, in order to alleviate some of the stress in piloting such a powerful mecha (although Lordgenome displayed the ability to singlehandedly control it during his reign, as he either killed or ejected anyone else on board during his takeover). One possible explanantion is in the fact that while Lordgenome fought the Anti-Spirals where they were, Simon relied on his consistant focus on one single being in the entire universe to compass his way to the Anti-Spiral home.
Similar to the Gurren Lagann itself, The Super Galaxy Gurren Lagann sports a colossal pair of sunglasses on its torso materialized during transformation along with a smaller pair for the head.The sheer power of this mecha shows it to be what is almost certainly the third most powerful machine in the series, capable of both deflecting and destroying objects, with Simon's piloting, on a planetary scale with ease. Like the Gurren Lagann, it also has the power to create drills from many points on its body at once and even uses a colossal version of its signature Giga Drill Break attack. Additionally, the mecha is equipped with thousands of cannons that appear via captain's command, from the treads of the mecha's drills (and, in some cases, within the threads of those cannons, which resembled drills), each capable of shooting a beam straight through the time-space continuum, used for hitting targets regardless of their locality and temporality in the universe.
In Gurren Lagann Parallel Works episode 8, which focused on Lord Genome's era as a Spiral Warrior, it is shown how he used the Cathedral Terra in Gunman mode to wipe out the human race. Though it resembled the Super Galaxy Gurren Lagann, it lacked the Simon shades and generally resembled Lazengann. It was piloted by Lazengann inside the Dekabutsu rather than the Gurren Lagann inside the Arc-Gurren Lagann. While in battleship mode, Cathedral Terra's abilities are even more destructive, able to take down several Ashtangas and their escorts in a single blast of its mighty cannon, albet with help from the surrounding ships. Also, it appears that the Cathedral Terra is a unique ship in the Spiral Warrior's arsenal, as no other ships of this class are ever seen. In 8, Lordgenome also demonstrated his ability to use a Giga Drill Break with Cathedral Terra, hinting that this attack is an innate skill of all spiral beings, rather than a "killing blow" of any particular individual as previously thought. This is evidenced by how Kittan's first and only attack involving spiral energy was the Giga Drill Break, while Simon's involved shooting his Gunmen through the ground and towards the sky.