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Dear Forum,

Luiz Figueiredo asked:

I have been trying to use the function Galois(p), for p polynomial to

get the Galois group of the splitting field of p. This function

returns an integer, the Galois type of the Galois group.

Can anyone tell me where there is a list of all Galois types for a

given degree, ie. which group correspond to which Galois types?

This is the index in the list of transitive groups. Thus if 'Galois' returns

'n' for an irreducible polynomial of degree 'd' then

'TransitiveGroup(d,n)' returns a permutation group which is permutation

isomorphic to the operation of the Galois group on the roots.

(The function works only for irreducible polynomials, accordingly the Galois

group acts transitively).

The arrangement is the same as in the article

@article{btlmck, author = "Gregory Butler and John McKay", title = "The transitive groups of degree up to 11", journal = CommAlg, volume = 11, year = 1983, pages = "863--911" }

There is no facility to deduce from this the actual action on the roots

when embedding the splitting field say in the complex numbers.

The groups obtained via 'TransitiveGroup' already have names in GAP (i.e.

TransitiveGroup(6,10) has the name "1/2[S(3)^2]2". These names however

are not yet 'cleaned up' and will change. They should not be used to refer

to these groups in the literature. (I have a newer list with

better names I can provide you with if you want to use names.)

(Your question seems to indicate that you still lack the manual section on

'GaloisType' and accordingly did not do the upgrade to version 3.4.2. As these

patches also fix at least one bug in the 'Galois' code you should seriously

consider upgrading.)

> Thanks,

You're welcome,

Alexander Hulpke

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