Assume I have simple group G and its cyclic Sylow subgroup H. Are there any nice properties of the induced representation ?

Any remarks "around the situation" (relaxing "simple" "cyclic" "Sylow") are also welcome.

Example: Consider GL_3(F_2) which is remarkbly isomorphic to PSL_2(F_7).

It has cyclic Sylow subgroups of order 7 and 3. And irreps with dimensions 1,3,3,6,7,8.

$Ind_{Sylow 7} = V_1 \oplus V_7\oplus 2V_8$

$Ind_{Sylow 3} = V_1 \oplus V_{3a}\oplus V_{3b} \oplus 2V_6\oplus 3 V_7 \oplus 2V_8$

Here induction is from trivial character.

This is result of numerical calcultion with character table (hope I did not make mistake).

Is there some advice which can help me to predict the results in advance at least partitialy ?

I am making similar calculations with other groups, using character is a little painful, if something can be done more easily - would be very helpful.

PS

Particular question: In particular I need to calculate inductions from non-trivial charcater in the cases above, any guess how to get the results without calculation ?