Possible Class equation for a group Determine the possible class equation for a group of order 21?
Until now I have found the following:
$1+3+3+7+7$
$1+1+1+3+3+3+9$
$1+1+1+1+1+1+1+7+7$
$1+1+1+1+1+1+1+1+1+3+3+3+3$
$1+1+1+\cdots +1 \ (21 \ \text{times})$
Is there any way to eliminate the choices from this equation? More importantly, how would we know that this is a complete list (Until now my attempt has just been guess and check after I found possible occurences of 1's) Is there any easier way to determine the class equation?

Given that |G|=21.Obviously,second,third and forth can be not possible class equations . Because from 1+1+1+1+1+1+1+7+7 ,we see that |Z(G)|=7 and |G/Z(G)|=3, which is prime.So G is abelien.Then we must have G=Z(G),which implies that |G|=|Z(G)| but |Z(G)|=7.
Again from 1+1+1+3+3+3+9,we see that |Z(G)|=3 and |G/Z(G)|=7, which is prime.So G is abelian.So we must have G=Z(G) and |G|=|Z(G)| but |Z(G)|=3.finally from 1+1+1+1+1+1+1+1+1+3+3+3+3,we see that |Z(G)|=9, also we know Z(G) is a subgroup of G and |G| is divisible by |Z(G)| but |Z(G)|=9 ,|G| is not divisible by |Z(G)|
