The name is evidently abbreviated, but the resolution is uncertain. A possibility is Primigenes (TM Nam 23229) or Primogenes (TM Nam 32894), under the interchange of g/k. In any case, ⲡⲣⲓⲙ- clearly points to a Latin origin of the name.
This seems to be the Egyptian common noun ⲡ-ⲧⲟⲟⲩ, 'mountain', used as a personal name. Such a name appears to be unattested in Coptic-Egyptian onomastics.
Meaning: perhaps ⲣ̄ⲧ-, 'potentiary?' (office) + -ⲛ (genitive) + ⲁⲥ-, 'daughter' + -ⲓ (epenthetic) = 'daughter of Irti' = 'daughter of potentiary'.
The phrase can be understood as both a compound anthroponym and an anonymous designation of the person, 'the daughter of the potentiary'.