What is meant by vector group for transformers and why is it important?

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What is vector group for transformers?
A vector group is the International Electrotechnical Commission (IEC) method of categorizing the high voltage (HV) windings and low voltage (LV) winding configurations of three-phase transformers.
The vector group designation indicates the windings configurations and the difference in phase angle between them.
Example: a wye HV winding and delta LV winding with a 30-degree lead is denoted as "Yd11".

Common vector group presentations are tabulated below.

Why is the vector group important to specify?
As different combinations of winding connections will result in different phase angles between the voltages on the windings, transformers connected in parallel must have the same vector group because the mismatching of phase angles will result in circulating current and other system disturbances.