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## Main Question or Discussion Point

Im doing a question where I have to calculate of composition of automorphisms of a cyclic p-group and something has got me confused. When constructing decompositions of cyclic groups I have gotten used to grouping the direct products of groups with orders of the same prime to a power e.g [itex]C_{20}\cong C_4\times C_5[/itex].

In this question however I have gotten to a stage where accorging to my lecturer [itex]Aut(C_8)\times Aut(C_9)\cong C_2\times C_2 \times C_6[/itex] and i don't understand why. I would have expressed it has [itex]C_4\times C_6[/itex] since there are 4 numbers less than 8 that are coprime to 8 (eulers totient function). Can anyone help clear up my confusion?

In this question however I have gotten to a stage where accorging to my lecturer [itex]Aut(C_8)\times Aut(C_9)\cong C_2\times C_2 \times C_6[/itex] and i don't understand why. I would have expressed it has [itex]C_4\times C_6[/itex] since there are 4 numbers less than 8 that are coprime to 8 (eulers totient function). Can anyone help clear up my confusion?