The Phoenix Conglomerate (sometimes called the Houo Conglomerate) is a large financial chain of corporations that houses and hides the interests of a clan of good shinobi. They are one of Hanzō Academy's primary investors.
When the head of the conglomerate's sole heir, Murasame, proved incapable of carrying the families shinobi legacy, they opted to adopted a distant relative who showed great promise. This ended up being Ikaruga. As a testament to this, she was given the family heirloom, Hien. This however lead to her earning the resentment of Murasame, as he truly wanted to carry out the family legacy, despite his lack of ability.
Rivalry with the Great Wolf Conglomerate
Because Ikaruga was adopted into the family, the Great Wolf conglomerate decided it needed to do the same in retaliation. This lead them to adopting Murakumo, who blamed Ikaruga for her being separated from her family.