Consider the incomplete character table for a group given below:&nbsp;(1)(1)(2)(2)(2)&nbsp;1abcdx111111x211−1−11x311−1−1−1x42−2000All the conjugacy classes are there.1) What is the order of the group?2) How many characters are missing?3) Find the missing character and complete the table.4) Find the order of the Kemel of the missing character.

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Consider the incomplete character table for a group given below:&amp;nbsp;(1)(1)(2)(2)(2)&amp;nbsp;1abcdx111111x211−1−11x311−1−1−1x42−2000All the conjugacy classes are there.1) What is the order of the group?2) How many characters are missing?3) Find the missing character and complete the table.4) Find the order of the Kemel of the missing character.

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1) The order of the group is expressed as rows×columns.The number of rows in the table is 4.The number of columns in the table is 5.Hence, order of the table is,order&amp;nbsp;=4×5=202)There is last row missing which make the table complete.There are 5 character in each row.So, the number of missing character is 5.3)The missing character are determined as,As known that,XG=∑i=15dixi=x1+x2+x3+x4+4x5x5=14(XG−x1−x2−x3−x4)Now,&amp;nbsp;x5(1)=4×1=4x5(a)=14×(0−1−1−1+2)=−14x5(b)=14×(0−1+1+1−0)=−14x5(c)=14×(0−1+1+1−0)=−14x5(d)=14×(0−1−1+1−0)=−14The complete table is,&amp;nbsp;1abcdx111111x211−1−11x311−1−1−1x42−2000x54−14−14−14−14&amp;nbsp;4) The order of the kernel of the missing character is 5.

Consider the incomplete character table for a group given below: \begin{array}{|c|c|}\hline&(1)&

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