I think some responses are being very hard on a three-year old who doesn't know what "hate" is. She obviously is fine with you and affectionate when it's the two of you.
And please realize it is VERY typical around this age for both girls and boys to favor one parent or the other. Usually they favor the parent who is around the least, the one who makes the big entrance at the end of each working day, over the parent who is around more of the time. This is normal and typical and is a phase -- a phase! That means the child will grow out of it, IF the adults treat it as no big deal and remind the child to be kind to the "other" parent and discipline appropriately for yelling and rudeness.
But is that happening in your case? Some posters are raging that your husband is egging her on, encouraging her, etc. Well, IS he? They are saying he's encouraging her but you, yourself, only say: "All this behavior thrills my husband because he claims she acts like this because I am never around." Does his being thrilled mean that he does indeed prompt her to love on him and be sassy and rude to you? Does he say and do things along those lines to make a joke out of her talking to you unacceptably? Does he laugh when she yells at you, for instance? It's hard to tell from the actual post though others seem to be assuming this is what he's doing.
IF he is egging her on: You and he need to have a serious talk. I worry for you that he's not going to have that talk because he believes you should be home with her and not working -- so the fault is all on you. Is that what's going on? Is that why he criticizes you for "never being around"? That's a bigger and harder issue between you and HIM -- not between you and a three-year-old. It sounds as if he's possibly using her to get your goat over another conflict where he wants you to stay home and you want/need to work. Hard to tell. Is that the larger problem? Does he also ding you for working out, or doing anything else? Does he leave you with her on weekends "because he 'has things to do'" (as you put it) and he treats you like a babysitter, and like he's doing you a huge favor just to spend weekend time with you both?
I see some red flags here that there are other issues going on besides your child's treatment of you. She is acting out what she's allowed to act out. You and your husband MUST be on the same page and have the same reactions EVERY time she sasses you or argues with you in his presence. He needs to tell her every single time, "You can't talk to mommy that way. When you talk to HER that way, it makes ME sad, and then she and I can't play with you/talk to you/be with you." And then she gets time out. Please look up Jo Frost's books on discipline and her time out principles (she was TV's "supernanny") and be very, very consistent -- daughter does not get out of time out if she gets up; if she begs daddy; if she yells at mommy (which makes the time out longer!!); etc.
Please update and let us know if there really is more to the picture here. Your child's actions are not acceptable but she also is three-barely out of babyhood and acting on what is permitted by him and by you as well. If the issue is that he's using her to make a point with you and to rub her affections for him in your face -- that's a real marriage issue, frankly, not an issue with your child. If this is not that severe, then you both need to start using age-appropriate discipline (which means not overly hard and not going on and on!) with her. But HE must institute it, not just you.