# What is the intuition that this theorem should give?

I am reading the direct products and here is a theorem that I could't understand its serve well:

A group of order 30 is ismorphic to one of

$$C_{30}, C_5\times D(3), C_3\times D(5)\qquad \qquad,$$ or $$\qquad D(15)$$

Which type of groups are $$C_{30}, C_5, C_3$$? And I also couldn't follow the reasoning in this example that the book gave after that theorem:

The group with presentation $$\big $$ is isomorphic to the direct product $$C_3\times D(5)$$

• $C_n$ denotes the cyclic group of order $n$. – Prasun Biswas Dec 21 '17 at 0:49
• You should start from recalling that $D_n$ is not an abelian group if $n\geq 3$, then compare the multiplication tables of $C_3\times D_5$ and the group presented by such relations (this is a pretty inefficient approach, indeed, but it surely works). – Jack D'Aurizio Dec 21 '17 at 0:49
• You can find more detail atlas of finite group:brauer.maths.qmul.ac.uk/Atlas – 1ENİGMA1 Dec 21 '17 at 6:29
• Your first question answer is there:math.stackexchange.com/questions/569226/… – 1ENİGMA1 Dec 21 '17 at 6:36