Synthetic data generation is integral to ML pipelines, e.g., to augment training data, replace sensitive information, and even to power advanced platforms like DeepSeek. While LLMs fine-tuned for synthetic data generation are gaining traction, synthetic table generation -- a critical data type in business and science -- remains under-explored compared to text and image synthesis. This paper shows that LLMs, whether used as-is or after traditional fine-tuning, are inadequate for generating synthetic tables. Their autoregressive nature, combined with random order permutation during fine-tuning, hampers the modeling of functional dependencies and prevents capturing conditional mixtures of distributions essential for real-world constraints. We demonstrate that making LLMs permutation-aware can mitigate these issues.