Select the three vowels, and arrange them in 3! = 6 ways (EAI), (EIA), (IEA), etc.
Then, arrange the four consonants in 4! = 24 ways.
Insert the group of vowels in five ways: either to the left or the right of the consonants, or in any of the three spaces between two consonants. (For instance, if the vowel-group is arranged as AEI, and the consonant-group is arranged as LDNG, the groups may be arranged in five ways as (AEILDNG, LAEIDNG, LDAEING, LDNAEIG, LDNGAEI.)
3! × 4! × 5 = 720