# D6

#### alexandrabel90

find all the subgroups of D6 that have order 6.

my book says that there should be 6 but i can only think of 1, which is the one consisting of all the rotations.

#### tonio

find all the subgroups of D6 that have order 6.

my book says that there should be 6 but i can only think of 1, which is the one consisting of all the rotations.

We have that $$\displaystyle D_6\cong C_2\times S_3$$ , $$\displaystyle C_2:=\{c\;;\;c^2=1\}$$ , so two of the subgroups of order 6 are $$\displaystyle 1\times S_3\cong S_3\,,\,\,<c>\times <(123)>\cong C_6$$ and, as far as I can tell, there's

only one subgroup more of order 6 which I leave to you to find.

Anyway, I can't see how that book says there are 6 sbgps. of order 6...what book is that?

Tonio