EDIT : Cleaned up answer, added more info.

20 is small enough that it is possible to find ALL the symmetric 6-valent graphs on 20 vertices.

In fact, all the vertex-transitive graphs up to 32 vertices of any valency are known!

(See http://symomega.wordpress.com/2012/02/27/there-are-677402-vertex-transitive-graphs-on-32-vertices/)

On this page:

mapleta.maths.uwa.edu.au/~gordon/trans

Gordon Royle has a bunch of files containing all the vertex-transitive graphs on up to 31 vertices in graph6 format. The files are split in different categories so, if you scroll down, you will find a file containing the connected 6-regular vertex-transitive graphs. 

I went ahead and checked Gordon's data. Out of the 80 connected 6-valent vertex-transitive graphs on 20 vertices, only 5 are also edge-transitive. (They are also arc-transitive).

One of them is a circulant, three are Cayley graphs on F20 (the Frobenius group of order 20) and the last one is not a Cayley graph.


Here is some info about the graphs.
The first line is the graph6 data (I can post the adjacency matrices if you prefer that), the second line is a rough description of the automorphism group.
The other lines should be self-explanatory.






SsaCBLYNAWEOP@Q@@_`CRCagoJ?Bf?B_w      
(10x2):D6  
Circulant       
Girth 4  
Chromatic number 3

SsaCBLYNBOI_[?I_Ao?[??Mk@VOBZ??^_           
S5x2  
Non-Cayley       
Girth 4  
Chromatic number 2

SsaCB|_WB?K?EKEKB@_oW@kKEooK]?K]?         
2^10:S5  
Cayley on F20    
Girth 4  
Chromatic number 3

S{aSQ\`GGhI?oE@OpGc\`_eIAgROgXQ_B{?           
S5x2  
Cayley on F20    
Girth 3  
Chromatic number 4

S~aKYPDHGqCQCbOWCAg_VAQ?CoGCKW?B{            
S5x2   
Cayley on F20    
Girth 3   
Chromatic number 4.

For group calculations, I used Magma. For chromatic numbers, I used Sage. The first one has a solvable group, so will not contain the groups you are interested in. All the others contain copies of A5.