We learn quite a lot about Antigone in this exchange between the sisters which is unheard by the chorus. Here are a few:
The girls are still suffering the effects of their father's shame--"pain...sorrow...suffering...dishonor."
Antigone is the one who knows the latest news of the town, and particulary the news of their brother's fate. Ismene has heard nothing.
Antigone understands what she's suggesting to her sister is a punishable offense, which is why she takes her outside the city gates to talk with her about it.
Antigone believes her uncle, the King, has made the decree against properly burying the "traitor" specifically against the two of them. Then she adds, "Against me." She is apparently the one who will care most about her brother's eternal destiny.
Antigone challenges her sister to get her to act; she is obviously the stronger of the two, and she knows her sister to be less resolute.
Antigone clearly wants her sister's help--she asks her pointedly several times: Will you help me?"
Antigone is fiercely loyal, obviously.
Antigone understands she is a person with rights, not just a woman dependent on the kindness and generaosity of others: "He has no right to keep me from my own."
Antigone is disgusted with her sister, who says she's only a woman and must obey the laws set forth by men.
Antigone is willing to die in the pursuit of theis cause: "And if I die for it, what happiness!"
Antigone dismisses Ismene's fears for her: "You need not fear for me. Fear for yourself." She obviously believes her sister's soul is in danger if she allows this injustice to happen to her brother.
Antigone doesn't care who finds out about her plans.
Antigone is determined: "When I have tried and failed, I shall have failed.