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The Church of Satan is an entity promoting an atheistic belief system that sees Satan as a powerful symbol of specific human traits, seeing humans as the real beast, and that they should embrace all of their animalistic nature. In the short life of here and now, it is our obligation to become our own gods and enjoy life to the fullest, while renouncing supernatural gods and the moral restraints that come with them. LaVey promoted the Seven Deadly Sins as good things that we should embrace. He also promoted being a murderous threat to anyone who would interfere with our free will. This is called LaVeyan Satanism.
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There is some confusion surrounding LaVeyan Satanists performing spell and ceremonial rituals as laid out in The Satanic Bible, written by LaVey. LaVey didn't believe that rituals accomplished anything tangible, but rather were necessary theatrics for psychological fulfillment in place of religion.
Even in light of The Satanic Bible being a heavy product of plagiarism, valid points to his philosophy surrounding natural law and survival of the fittest have been embraced by many. Regardless, the book is a short read, and worth reading, even for those who are religious and afraid that reading it will make Satan eat their souls.
The CoS originally sought to promote political advocacy through its members, however it has consistently failed to do so because of its own inaction. Instead it had only served as a source of media attention during the years LaVey was still alive, with him and affiliates being interviewed on various shows talking about their beliefs. To date, they haven't seemed to have done much advocating, and the organization has lost members to starting their own groups over the years.
Groups risen from leaving the CoS over disputes and ideological differences include the Temple of Set. LaVey's death led to infighting in the CoS. Blanche Barton, LaVey's partner, became the new head of the organization in accordance with his living will. Karla LaVey, Anton's daughter, was strongly against this, wanting to head the CoS herself. Being unable to do so, she went on to found the First Satanic Church.
Currently, the number of members of the CoS is unknown and never released. It is believed to be only in the hundreds despite its popularity and unaffiliated individuals calling themselves LaVeyan Satanists.