Maybe better draw a quick genealogy (basing on Svetonius)...

The correct order of the Caesari (Emperiors) is:

Augustus > Tiberius > Caligula > Claudius > Nero.



Germanicus, son of Drusus Claudius and Antonia, was adopted by Tiberius.

Germanicus died at Anthiochia after a long sick, Tiberius was suspected to be the instigator of a possible poisoning.

Wife of Germanicus was Agrippina, daughter of Marcus Agrippa and Iulia. Of their sons survived three males and three females: Nero, Drusus, Caius, Agrippina, Drusilla and Livilla.

Tiberius made the Senatus decree hostis both Nero and Drusus, so the only remaining son of Germanicus was Caius.
Being born and raised between legionaries, he received a cognomen from the soldiers: Caligula (the little caliga, the soldiers shoe).

So now we can try to write a full name for Caligula Caesar:

Caius Iulius Germanicus filius Drusus nepos Octavianus pronepos (tribe?) Caligula

[I doubt Caligula took Tiberius in his name since it is possible he killed Tiberius]



Son of Drusus was also Tiberius Claudius Drusus.
When Tiberius (the emperior) adopted Germanicus, Agustus ordered him to adopt also Claudius (the last of gens Claudia).
While Germanicus took the nomen of gens Iulia, Tiberius Claudius kept his oroginal one but took the name of his brother as cognomen bocoming: Tiberius Claudius Germanicus (caligula was his nephew).

So now we can try to write a full name for Claudius Caesar:

Tiberius Claudius Drusus filius Octavianus nepos (unkwn?) pronepos (tribe?) Germanicus



Nero is son of Domitius and Agrippina (daughter of Germanicus).
Nero is the nomen Caligula choosen for his nephew.
Later Nero was adopted by Claudius.

So now we can try to write a full name for Nero Caesar:

Nero Claudius Domitius filius Caius nepos Drusus pronepos (tribe?) ?

[the only possible cognomen I found for Nero is Enobarbus, as his brother Britannicus was used to call him, also this was one of the cognomini from his father' gens].