The possible dimension sets are \((1,1,24),(1,2,12),(1,3,8),(1,4,6),(2,2,6),(2,3,4)\).

Using \(2(lw+lh+wh)\), their surface areas are respectively \(98,76,70,68,56,52\text{ cm}^2\).

The smallest is fifty-two, for two by three by four.

A shortest edge of three would require volume at least twenty-seven, so the listed cases with shortest edge one or two are exhaustive.

Answer

Two by three by four centimetres, with surface area \(52\text{ cm}^2\).