[GAP Forum] transitive groups with irreducible representations defined over QQ[sqrt(5)]

[Vincent Delecroix]

Dear all,

I am looking for transitive groups G of S_d (I don't know d) with
the following properties

  (*1) G admits an irreducible representation defined over
       QQ[sqrt(5)]  (in particular the character is real)

  (*2) the stabilizer H of 1 in G admits invariant vectors
      in that irreducible representation.

Question number one: knowing an irreducible representation defined
over QQ[sqrt(5)] what is the fastest way to check for (*2)?

Question number 2: so far, I am computing the full character table
for the group G and this is very expensive. Do you know of any
practicable criterion that would allow me to discard groups without
property (*1)?

Question number 3: given G is there a way to access the characters
defined over QQ[E(5)] but not the one with higher conductors without
filtering the list of irreducible characters?

Best
Vincent