Non-Faithful Connected Quandles of Order at most 47 that are Conjugation Quandles
                       


There are 790 connected quandles in the GAP package Rig written by Leandro Vendramin.
These comprise all connected quandles of order at most 47.

We are interested here in determining which of these are conjugation quandles. 
Say that a quandle X is a conjugation quandle if it is isomorphic to a conjugacy
class C of some group G furnished with the quandle operation of conjugation. 
In this case we say that X is a conjugation quandle on G. 

Note that if X is a subquandle of G with the operation of conjugation and X is a connected 
quandle then X is a conjugacy class in the subgroup <X> of G generated by X.

It is known that if a quandle X is faithful then it is a conjugation quandle on the
group Inn(X). It is easy to check that of the 790 Rig quandles only the following
66 quandles are not faithful. Here RIG[i] is the i-th Rig quandle and Q(n,j) is
the j-th Rig quandle of order n. In the Rig package Q(n,j) is denoted by 
SmallQuandle(n,j).  Here is a list using this notation of the 66 non-faithful
Rig quandles:

1) RIG[13] = Q(8,1) 
2) RIG[34] = Q(12,1) 
3) RIG[35] = Q(12,2) 
4) RIG[117] = Q(20,3) 
5) RIG[155] = Q(24,1) 
6) RIG[156] = Q(24,2) 
7) RIG[157] = Q(24,3) 
8) RIG[158] = Q(24,4) 
9) RIG[159] = Q(24,5) 
10) RIG[160] = Q(24,6) 
11) RIG[161] = Q(24,7) 
12) RIG[162] = Q(24,8) 
13) RIG[168] = Q(24,14) 
14) RIG[169] = Q(24,15) 
15) RIG[170] = Q(24,16) 
16) RIG[171] = Q(24,17) 
17) RIG[172] = Q(24,18) 
18) RIG[173] = Q(24,19) 
19) RIG[174] = Q(24,20) 
20) RIG[231] = Q(27,1) 
21) RIG[236] = Q(27,6) 
22) RIG[244] = Q(27,14) 
23) RIG[336] = Q(30,1) 
24) RIG[339] = Q(30,4) 
25) RIG[340] = Q(30,5) 
26) RIG[351] = Q(30,16) 
27) RIG[389] = Q(32,1) 
28) RIG[390] = Q(32,2) 
29) RIG[391] = Q(32,3) 
30) RIG[392] = Q(32,4) 
31) RIG[393] = Q(32,5) 
32) RIG[394] = Q(32,6) 
33) RIG[395] = Q(32,7) 
34) RIG[396] = Q(32,8) 
35) RIG[397] = Q(32,9) 
36) RIG[432] = Q(36,1) 
37) RIG[435] = Q(36,4) 
38) RIG[438] = Q(36,7) 
39) RIG[440] = Q(36,9) 
40) RIG[442] = Q(36,11) 
41) RIG[444] = Q(36,13) 
42) RIG[445] = Q(36,14) 
43) RIG[448] = Q(36,17) 
44) RIG[451] = Q(36,20) 
45) RIG[452] = Q(36,21) 
46) RIG[459] = Q(36,28) 
47) RIG[461] = Q(36,30) 
48) RIG[488] = Q(36,57) 
49) RIG[489] = Q(36,58) 
50) RIG[518] = Q(40,1) 
51) RIG[519] = Q(40,2) 
52) RIG[520] = Q(40,3) 
53) RIG[521] = Q(40,4) 
54) RIG[522] = Q(40,5) 
55) RIG[523] = Q(40,6) 
56) RIG[529] = Q(40,12) 
57) RIG[530] = Q(40,13) 
58) RIG[534] = Q(40,17) 
59) RIG[535] = Q(40,18) 
60) RIG[536] = Q(40,19) 
61) RIG[537] = Q(40,20) 
62) RIG[538] = Q(40,21) 
63) RIG[562] = Q(42,12) 
64) RIG[571] = Q(42,21) 
65) RIG[574] = Q(42,24) 
66) RIG[614] = Q(45,29) 

Using the following GAP/Rig function suggested by Leandro Vendramin
we determined which of the above 66 quandles are conjugation quandles.
It turns out that exactly 30 of them are conjugation quandles.

If Q is a conjugation quandle then this procedure returns the finite
enveloping group (for which Q is a conjugation quandle), otherwise
it returns fail. 

IsConjugationQuandle:=function(Q)
  local G, F;
  G:=FiniteEnvelopingGroup(Q);
  F:=GeneratorsOfGroup(G);
  if Size(F) = Size(Q) then 
     return G; 
  else
     return fail;
  fi;
end;

We find that the following 30 non-faithful quandles are conjugation quandles.
In most cases we were able to identify the finite enveloping group using GAP's function
IdGroup. In some cases we were able to find a smaller group than the finite enveloping
group by random searching. In case we could not identify the finite enveloping group
(or find a smaller group) in this list we just state its order.


1) RIG[34] = Q(12,1) on groups SmallGroup(48,29)
2) RIG[117] = Q(20,3) on groups SmallGroup(240,90)
3) RIG[157] = Q(24,3) on groups SmallGroup(192,181)
4) RIG[158] = Q(24,4) on groups SmallGroup(384,572)
5) RIG[159] = Q(24,5) on groups SmallGroup(384,18127)
6) RIG[160] = Q(24,6) on groups SmallGroup(192,1489)
7) RIG[168] = Q(24,14) on groups SmallGroup(384,570)
8) RIG[170] = Q(24,16) on groups SmallGroup(384,18133)
9) RIG[171] = Q(24,17) on groups SmallGroup(384,18130)
10) RIG[336] = Q(30,1) on groups SmallGroup(120,5)
11) RIG[351] = Q(30,16) on groups SmallGroup(1440,4591)
12) RIG[393] = Q(32,5) on groups SmallGroup(384,611)
13) RIG[394] = Q(32,6) on groups SmallGroup(384,618)
14) RIG[395] = Q(32,7) on groups SmallGroup(384,613)
15) RIG[396] = Q(32,8) on groups SmallGroup(384,613)
16) RIG[432] = Q(36,1) on groups SmallGroup(432,258)
17) RIG[435] = Q(36,4) on groups SmallGroup(144,32)
18) RIG[448] = Q(36,17) on groups SmallGroup(1296,697)
19) RIG[451] = Q(36,20) on groups SmallGroup(432,245)
20) RIG[452] = Q(36,21) on groups SmallGroup(432,245)
21) RIG[459] = Q(36,28) on groups SmallGroup(1296,3085)
22) RIG[461] = Q(36,30), Order of Finite Enveloping Group = 3888)
23) RIG[529] = Q(40,12), Order of Finite Enveloping Group = 3840)
24) RIG[530] = Q(40,13), Order of Finite Enveloping Group = 3840)
25) RIG[534] = Q(40,17), Order of Finite Enveloping Group = 15360)
26) RIG[535] = Q(40,18), Order of Finite Enveloping Group = 3840)
27) RIG[536] = Q(40,19) on groups SmallGroup(1280,1116364)
28) RIG[537] = Q(40,20) on groups SmallGroup(320,1582)
29) RIG[562] = Q(42,12) on groups SmallGroup(672,1047)
30) RIG[571] = Q(42,21), Order of Finite Enveloping Group = 10080)

Here's a GAP readable list for the above data on conjugation quandles:
CQ:=[
[ 34, [ 12, 1 ], [ 48, 29 ] ],
[ 117, [ 20, 3 ], [ 240, 90 ] ],
[ 157, [ 24, 3 ], [ 192, 181 ] ],
[ 158, [ 24, 4 ], [ 384, 572 ] ],
[ 159, [ 24, 5 ], [ 384, 18127 ] ],
[ 160, [ 24, 6 ], [ 192, 1489 ] ],
[ 168, [ 24, 14 ], [ 384, 570 ] ],
[ 170, [ 24, 16 ], [ 384, 18133 ] ],
[ 171, [ 24, 17 ], [ 384, 18130 ] ],
[ 336, [ 30, 1 ], [ 120,5 ] ],
[ 351, [ 30, 16 ], [ 1440, 4591 ] ],
[ 393, [ 32, 5 ], [ 384,611 ] ],
[ 394, [ 32, 6 ], [384, 618 ] ],
[ 395, [ 32, 7 ], [ 384, 613 ] ],
[ 396, [ 32, 8 ], [ 384, 613 ] ],
[ 432, [ 36, 1 ], [ 432, 258 ] ],
[ 435, [ 36, 4 ], [ 144, 32 ] ],
[ 448, [ 36, 17 ], [ 1296, 697 ] ],
[ 451, [ 36, 20 ], [ 432, 245 ] ],
[ 452, [ 36, 21 ], [ 432, 245 ] ],
[ 459, [ 36, 28 ], [ 1296, 3085 ] ],
[ 461, [ 36, 30 ], 3888 ],
[ 529, [ 40, 12 ], 3840 ],
[ 530, [ 40, 13 ], 3840 ],
[ 534, [ 40, 17 ], 15360 ],
[ 535, [ 40, 18 ], 3840 ],
[ 536, [ 40, 19 ], [ 1280, 1116364 ] ],
[ 537, [ 40, 20 ], [ 320, 1582 ] ],
[ 562, [ 42, 12 ], [ 672, 1047 ] ],
[ 571, [ 42, 21 ], 10080 ]
];;


The following are the 36 non-faithful connected quandles of order < 48
that are not conjugation quandles. The number at the end of each line
is Size(Q)/Size(inn(Q)), where inn:Q -> Inn(Q) is the right translation
mapping.

1) RIG[13] = Q(8,1), 2 
2) RIG[35] = Q(12,2), 2 
3) RIG[155] = Q(24,1), 2 
4) RIG[156] = Q(24,2), 4 
5) RIG[161] = Q(24,7), 2 
6) RIG[162] = Q(24,8), 2 
7) RIG[169] = Q(24,15), 2 
8) RIG[172] = Q(24,18), 2 
9) RIG[173] = Q(24,19), 2 
10) RIG[174] = Q(24,20), 2 
11) RIG[231] = Q(27,1), 3 
12) RIG[236] = Q(27,6), 3 
13) RIG[244] = Q(27,14), 3 
14) RIG[339] = Q(30,4), 3 
15) RIG[340] = Q(30,5), 2 
16) RIG[389] = Q(32,1), 2 
17) RIG[390] = Q(32,2), 2 
18) RIG[391] = Q(32,3), 2 
19) RIG[392] = Q(32,4), 2 
20) RIG[397] = Q(32,9), 2 
21) RIG[438] = Q(36,7), 2 
22) RIG[440] = Q(36,9), 2 
23) RIG[442] = Q(36,11), 2 
24) RIG[444] = Q(36,13), 2 
25) RIG[445] = Q(36,14), 2 
26) RIG[488] = Q(36,57), 3 
27) RIG[489] = Q(36,58), 3 
28) RIG[518] = Q(40,1), 2 
29) RIG[519] = Q(40,2), 2 
30) RIG[520] = Q(40,3), 2 
31) RIG[521] = Q(40,4), 2 
32) RIG[522] = Q(40,5), 2 
33) RIG[523] = Q(40,6), 2 
34) RIG[538] = Q(40,21), 2 
35) RIG[574] = Q(42,24), 2 
36) RIG[614] = Q(45,29), 3