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Dover's players were subjected to racial abuse during their National League win at Hartlepool on Saturday.
Whites boss Andy Hessenthaler said it was a "sad day for football".
The game was halted for several minutes after Dover striker Inih Effiong scored a 36th-minute penalty to put his team ahead.
Whites players celebrated in front of the home supporters, where racist comments were aimed at Dover players by some fans.
Hessenthaler told reporters after the game that he had asked his players if they wanted to come off but Dover's players opted to stay on.
It was Hartlepool who lost their discipline, with two home players collecting red cards during the game as they finished with nine men.
Dover's players have spoken of their disgust on Twitter this evening.
Jai Reason said: "I’ve never been involved in a game like that, so proud to be part of this team how we all dealt with the racism and monkey chants today from the home supporters.
"These people should not be anywhere near a football ground, more needs to be done."
Ricky Modeste wrote: "What a sad day for football. I’m all for fans giving opposition players stick but the disgraceful racist comments and gestures that us players received today was way over the line.
"We are only human and are there to do a job."
Dover assistant manager Nicky Southall, himself a former Hartlepool player, said: "I have to say being a former player for four years, I'm absolutely shocked and disappointed by the actions of some individuals (at Hartlepool). No place for this... sad day for football."
Hartlepool manager Craig Hignett said: "Racism has no place in football and no place in life and those people will be dealt with severely because that's not us as a Football Club."
Hartlepool chief excutive Mark Maguire said: “I can honestly say that I have never witnessed a game of football like that, but it is important that we separate our reaction to the game, and the officials, to the events following Dover’s first goal.
“Whatever the rights and wrongs of the decision, or the aftermath to the penalty which was scored, Hartlepool United utterly condemn racist behaviour of any description.
“We are a proudly inclusive as a Club, and we know that the vast majority of Pools fans will join us in standing up against words and actions which have no place in football or society.
“As I understand it, in this, the early stages after the event, individuals have been identified and will be dealt with in the strongest possible manner by both the Club and police.
“Equally, I personally spoke to the people on the receiving end of the abuse, assuring them that we would take the necessary action.
“We also made a public announcement at half time, setting out our condemnation of such behaviour, whilst I also spoke with the Chairman of Dover to offer my apology.
“As a Club we will protect the principles held by the vast majority of our fan base who would be insulted by their reputation being tarnished in this manner. In these type of circumstances actions speak louder than words and we will be judged accordingly.
“On days like today, Directors, Management and staff of both clubs have a responsibility to stand up and be counted, and we have no hesitation in reacting in the strongest possible manner.
“Of a completely separate nature is any concern surrounding events on and off the pitch during and immediately after the game, and we will deal with this accordingly with the appropriate authorities.”