Sphynx

Sphinx (スフィンクス) is the pet calico cat of Index. First found by Index in the second volume of the Light Novels as an abandoned cat, it later becomes a constant companion to Index with one not often appearing without the other.

Appearance
Sphinx is a kitten, and as such is small and can be easily carried around by Index. In fact it is often seen inside of Index's Walking Church. Sphinx, like cats of its breed, is covered with patches of brown and black fur covering its white coat. Sphinx has yellow-green eyes and sports a red collar around its neck (after being taken in by Index).

Himegami Aisa: Deep Blood
Sphinx is first seen by Index inside a box after Touma and she finished eating in Macronall Hamburger and meeting Himegami Aisa. Index wanted to take in Sphinx, however, Touma rejects her request as it is against the rules in his dormitory to keep pets. Because of their arguing, Sphinx runs away, with Index trying to follow, however, she quickly discovers a mana source set up by Stiyl and runs off to investigate, leaving Touma alone who later meets Stiyl Magnus for the first time after the Johann's Pen Incident.

When Index and Touma come home, he discovers that Index has taken in Sphinx. Though reluctant at first, Touma allows Index to keep Sphinx and shortly leaves for Misawa Cram School with Stiyl. Index later takes a bath with Sphinx, and becoming suspicious of Touma's actions decides to find him, she eventually follows Stiyl's mana trail to Misawa Cram School, where she meets Aureolus Izzard, who makes her fall asleep.

Touma later realizes that Index may be in danger, as he discovers Sphinx with Index's hood near the school.

By the end of the arc, Index pays a visit to Touma in a certain hospital, bringing Sphinx along where it proceeds to scratch Touma's face.

Sisters Arc
In the time between the Deep Blood Arc and the Sisters Arc, Sphinx somehow acquired fleas, which lead Index and Aisa to try find ways to get rid of it. When Touma comes home to his dormitory with Misaka 10032, he discovers that Index is going to use sage as away to remove the fleas, while Aisa decided that she was going to use an insecticide spray to purge Sphinx of the fleas. Sphinx is later spared of their remedies when Misaka 10032 decides to intervene, by giving Sphinx a slight electric shock using her abilities to kill the fleas. The scene is slightly different in the manga version (since it skips the Deep Blood Arc), where Index just recently acquired Sphinx and Touma as well to his surprise, however, Misaka 10032 still shocks Sphinx in the end, with Index being highly impressed of her.

Three Stories Arc
The majority of the arc has Sphinx with Index, however, when she was kidnapped by Yamisaka Ouma, Sphinx was carried around by Touma until he saved Index. When Yamisaka after receiving serious injuries from Index's grimoire he was about to give up when Touma told Sphinx to scratch Yamisaka in the face.

Kazakiri Hyouka and the Imaginary Number Sector
Index carries Sphinx with her to find Touma, as she had no lunch. After meeting Kazakiri Hyouka and reuniting with Touma, Index allows Hyouka to play with Sphinx, she tries to hug Sphinx but is suffocated by her large breast size, as such Sphinx is deterred by her, to the lament of Hyouka.

Later, after being teleported by Shirai Kuroko when Sphinx was with Index. Sphinx runs away for some reason, forcing Index chase after Sphinx, because of this Sherry Cromwell's golem finds Index and attacks her.

In the anime ending of the 1st season, Sphinx mimics Index's facial expression when she discovers that Touma is eating something, when she was starving. Index later bites Touma as well as Sphinx.

The Book of the Law Arc
Sphinx is in the care of Maika Tsuchimikado after Stiyl 'kidnaps' Index to force Touma to get involved in his current mission to retrieve Orsola Aquinas and the Book of the Law.

UK Civil War arc
Sphinx was sent ahead on the supersonic plane to UK, and was taken in by Itsuwa who arrived to try and pick up Touma and Index but ended up with only Sphinx. Later, before Touma and company headed off to fight against Carrisa in the climax of the civil war, Sphinx was seen trying to snatch food in the from various other nuns. After Touma and others have left, Rimea arrived at the place where Touma was, and took a liking to Sphinx and considered taking the cat in, much to her loyal pet dog's displeasure.

Sphinx was last seen with Rimea in the light novels, as of Volume 20. Presumbly Rimea have taken Sphinx in temporarily after its owners (Touma and Index) are busy elsewhere.