Quartic symmetric and semisymmetric G-graphs

We have introduce a new type of graphs from a group called G-graphs, these graphs give us a lot of information about the structure of groups. Moreover, these type of graphs can be used in any area of sciences where CAYLEY graphs occurs. G-graphs are very efficient to construct symmetric and semisymmetric graphs. Indeed we have built a power full algorithm to construct symmetric and semisymmetric graphs. The following table has been built in 38 minutes on a 2 gigahertz Athlon (32 bit processor).
In green are semisymmetric G-graphs, the others are symmetric.

O(Gamma)O(G)GNauty KeyAutomorphism groupSimple
816sg-16-2
sg-16-3
sg-16-4
[ 1bec 36a0a8e d566c4c]1 orbit; group size=1152;Yes
1020sg-20-3[ 197ec acba481 527bf19]1 orbit; group size=240;Yes
1224sg-24-12[ 107ec 3785a74 4cbbaed]1 orbit; group size=768;Yes
1632sg-32-2
sg-32-10
sg-32-11
sg-32-13
[ 2389ec 761d8d3 9b71d9a]1 orbit; group size=4096;Yes
1632sg-32-6[ 2bb8ec 23a2f94 4eefea5]1 orbit; group size=384;Yes
1836sg-36-9[fdc1acb ea9fdcb 70f0b7a]1 orbit; group size=144;Yes
2040sg-40-12[b12e2e0 8cbf445 7d9f1d4]1 orbit; group size=80;Yes
2448sg-48-11
sg-48-12
sg-48-19
sg-48-48
[e18db91 b870c97 a8b037f]1 orbit; group size=98304;Yes
2448sg-48-30[33a2afc ffb72ec ab7d92d]2 orbits; group size=3072;Yes
2652sg-52-3[b27d6b9 4981e0f ff23b0f]1 orbit; group size=104;Yes
3060sg-60-7[ 8dcddd a885ace ba0156d]1 orbit; group size=720;Yes
3264sg-64-18[865a651 8f844e4 f22c737]1 orbit; group size=4096;Yes
3264sg-64-34[e0299e1 22ece48 88cef4]1 orbit; group size=256;Yes
3264sg-64-8
sg-64-21
sg-64-39
sg-64-41
sg-64-48
[39a0d28 34b79b2 28b5f1b]1 orbit; group size=2097152;Yes
3264sg-64-9
sg-64-20
sg-64-23
sg-64-32
sg-64-33
sg-64-35
[c058a3d f9465e1 6309279]2 orbits; group size=294912;Yes
3468sg-68-3[b8319c9 1c8ecd4 703be37]1 orbit; group size=136;Yes
3672sg-72-15[b1e7130 7f5af6b 9c6aaca]1 orbit; group size=9437184;Yes
3672sg-72-41[eb4a1c6 b1e575d f4d071b]1 orbit; group size=144;Yes
3672sg-72-45[626f92a a310a6c 20caa1a]1 orbit; group size=288;Yes
4080sg-80-11
sg-80-12
sg-80-19
[ 29ce33 bf2fc2e a04236a]1 orbit; group size=41943040;Yes
4080sg-80-30[abd426c 801afeb 70a9c2f]1 orbit; group size=160;Yes
4080sg-80-31[4208c34 225ec90 3d0f4b7]1 orbit; group size=160;Yes
4080sg-80-34[e6bab40 3d403e8 9168d21]1 orbit; group size=160;Yes
4896sg-96-13
sg-96-23
sg-96-24
sg-96-32
sg-96-38
sg-96-187
[89fd128 6c1a151 c996ef3]1 orbit; group size=805306368;Yes
4896sg-96-185
sg-96-186
sg-96-194
sg-96-195
[ddfde41 25422e1 76634f5]2 orbits; group size=3145728;Yes
4896sg-96-227[7ad40f3 f20f82a 481ce5]1 orbit; group size=768;Yes
4896sg-96-41[24f9da7 6d7e3d6 2a795f1]1 orbit; group size=98304;Yes
4896sg-96-67[7cfeeaf 764531b 5e79462]1 orbit; group size=384;Yes
50100sg-100-10[a9a44c5 6c9d569 3178280]1 orbit; group size=200;Yes
50100sg-100-12[1a3d9e1 233229c 6db4d3a]1 orbit; group size=400;Yes
50100sg-100-3[4763b37 9c095d 4da592c]1 orbit; group size=200;Yes
52104sg-104-12[9a33b70 9c7132f ad6e409]1 orbit; group size=208;Yes
54108sg-108-15[c3f8fab c8e6f9d a11b82d]1 orbit; group size=432;Yes
54108sg-108-37[9e80297 9746eee 2920284]1 orbit; group size=1296;Yes
56112sg-112-10
sg-112-11
sg-112-18
[8d3a57f b88fb1f 5c52d54]1 orbit; group size=1.5032385536e10;Yes
58116sg-116-3[aec78cc 241d052 df8460c]1 orbit; group size=232;Yes
60120sg-120-34[5f68fa2 44bf120 4daaf34]1 orbit; group size=480;Yes
60120sg-120-38[ 420015 177d68 2d5d8bb]1 orbit; group size=6.4424509440e10;Yes
60120sg-120-41[a53840c 9bf0f60 c561c97]1 orbit; group size=240;Yes
64128sg-128-122
sg-128-134
sg-128-136
sg-128-139
sg-128-141
[720e9a0 8d596e8 edbc284]2 orbits; group size=25165824;Yes
64128sg-128-123[2047b22 6452816 57caf32]1 orbit; group size=512;Yes
64128sg-128-140[6de82ec 372ac90 1489f84]1 orbit; group size=512;Yes
64128sg-128-144
sg-128-146
[db22e7d ad3904b 47b7da4]2 orbits; group size=2048;Yes
64128sg-128-16
sg-128-29
sg-128-36
sg-128-83
sg-128-84
sg-128-125
[ccbf168 647cca 65cfcf4]1 orbit; group size=4096;Yes
64128sg-128-21
sg-128-116
[5c22c75 5b6784 df5f16c]1 orbit; group size=2097152; 1 10 Yes
64128sg-128-26
sg-128-71
sg-128-72
sg-128-75
sg-128-76
sg-128-80
sg-128-118
[b47ae4f e30ffd5 ef7e64]2 orbits; group size=268435456;Yes
64128sg-128-79
sg-128-112
sg-128-148
sg-128-150
sg-128-156
[2aeeb58 8a79685 9109172]1 orbit; group size=2.7487790694e11;Yes
68136sg-136-13[f7d83ca c58807b 22a8f4a]1 orbit; group size=272;Yes
70140sg-140-6[ 145b91 116c5a8 26db4]1 orbit; group size=280;Yes
72144sg-144-11
sg-144-12
sg-144-19
sg-144-109
[ 5347b0 7ba3880 dd4b03c]1 orbit; group size=4.9478023250e12;Yes
72144sg-144-115
sg-144-116
[4459ecf f4e0c54 f6ded4f]2 orbits; group size=37748736;Yes
72144sg-144-120[3034fca 6a2d429 b9025da]2 orbits; group size=144;Yes
72144sg-144-132[196cd50 81d9c1f f227174]1 orbit; group size=576;Yes
72144sg-144-133[5e1b0c7 9bdb63d db0147e]1 orbit; group size=576;Yes
72144sg-144-136[e6ccc97 56302fe e7cb151]1 orbit; group size=576;Yes
72144sg-144-33[137deb7 e3ee03d 44836e2]2 orbits; group size=2415919104;Yes
74148sg-148-3[36f79dd e1e2bb2 8efe195]1 orbit; group size=296;Yes
78156sg-156-10[ba5d000 5f1996b fdbb9c9]1 orbit; group size=312;Yes
80160sg-160-13
sg-160-23
sg-160-24
sg-160-32
sg-160-38
[8185ebb fabe6a3 ddefee2]1 orbit; group size=8.7960930222e13;Yes
80160sg-160-234[9528797 4c6c3ca 12a97d4]2 orbits; group size=335544320;Yes
80160sg-160-41[c039596 1e41f50 6416eb2]1 orbit; group size=41943040;Yes
80160sg-160-68[7002bdd f63dac3 e1a99c]1 orbit; group size=320;Yes
80160sg-160-69[7be32d6 1caff41 6add779]1 orbit; group size=320;Yes
80160sg-160-74[732ac42 2c47d25 713ba7a]1 orbit; group size=320;Yes
80160sg-160-81[961719e 1ad3de0 f5a3629]1 orbit; group size=320;Yes
80160sg-160-83
sg-160-84
[6221b7a 61d0e8c 244a101]2 orbits; group size=83886080;Yes
80160sg-160-86[c54bc9f 466f8fa 8b4047]1 orbit; group size=320;Yes
82164sg-164-3[3e9e636 e2a19f8 bf04140]1 orbit; group size=328;Yes
84168sg-168-42[77a6a75 b27c238 cbf2152]1 orbit; group size=672;Yes
84168sg-168-46[9b428eb 2efd8fa 9842b07]1 orbit; group size=3.6943590693e14;Yes
88176sg-176-10
sg-176-11
sg-176-18
[dba8cfb ed026d d9046b0]1 orbit; group size=1.5481123719e15;Yes
90180sg-180-24[cfc0da3 688ead9 1d5edd8]1 orbit; group size=720;Yes
90180sg-180-25[e8c7524 696f9d5 fd3ddec]1 orbit; group size=360;Yes
90180sg-180-6[f9543d8 9954efe 4295e70]1 orbit; group size=360;Yes
96192sg-192-100[ba4a49c 942519b c0fd5df]1 orbit; group size=768;Yes
96192sg-192-104[87d055a 19f5348 584884]1 orbit; group size=768;Yes
96192sg-192-1491[417d41c 386268b ede2a2a]1 orbit; group size=1536;Yes
96192sg-192-1493[2bf2242 e18a4d2 38b5fd0]1 orbit; group size=768;Yes
96192sg-192-1495[6cc3de2 73c7799 c0e79b0]1 orbit; group size=768;Yes
96192sg-192-1538[ededeb5 d5895e7 d45e489]1 orbit; group size=98304; Yes
96192sg-192-185[3dabc11 c438fc9 93ec60f]2 orbits; group size=192;Yes
96192sg-192-26
sg-192-32
sg-192-33
sg-192-34
sg-192-35
sg-192-86
sg-192-114
sg-192-957
sg-192-960
sg-192-971
sg-192-972
[fcd2c4e 5a006e2 d3a8d28]2 orbits; group size=1.6492674417e12;Yes
96192sg-192-29
sg-192-62
sg-192-64
sg-192-77
sg-192-110
sg-192-961
[2e35451 a6c85da a2f0d91]1 orbit; group size=2.7021597764e16;Yes
96192sg-192-85
sg-192-116
[5653297 8804ea9 a64d065]1 orbit; group size=805306368;Yes
96192sg-192-955
sg-192-963
sg-192-964
sg-192-969
sg-192-970
sg-192-987
sg-192-989
sg-192-991
[371b148 8f55f06 dbaece4]2 orbits; group size=1.2884901888e10;Yes
96192sg-192-981[100d0a2 a2ac7ee e2520c9]1 orbit; group size=768;Yes
96192sg-192-982[f97d8f7 b9c3e0d 769a29c]1 orbit; group size=768;Yes
98196sg-196-8[8f20346 cd3605e 493641a]1 orbit; group size=784;Yes
100200sg-200-12[e843202 b21e109 5674947]1 orbit; group size=400;Yes
100200sg-200-42[a61b969 f95de98 bdb3b73]2 orbits; group size=400;Yes
100200sg-200-44[7dc484e 38e5966 b1c2060]1 orbit; group size=800;Yes
100200sg-200-46[4ec49f8 91ecad7 5e45595]1 orbit; group size=400;Yes
100200sg-200-48[10c8431 a1d7443 c0d0ca8]1 orbit; group size=800;Yes
102204sg-204-6[c12ff57 d52ad23 d26b0b4]1 orbit; group size=408;Yes
104208sg-208-11
sg-208-12
sg-208-19
[b88c91c 6ea041b 2d1128a]1 orbit; group size=4.6837436125e17;Yes
104208sg-208-30[c4a8bec 556762f 4dacfc3]1 orbit; group size=416;Yes
104208sg-208-31[ fba0c0 d34eaba df840c2]1 orbit; group size=416;Yes
104208sg-208-34[2fb929e 2343951 59661ec]1 orbit; group size=416;Yes
106212sg-212-3[d391af6 8f4eb90 8558e97]1 orbit; group size=424;Yes
108216sg-216-100[55bdafa 40a336b d3dbe9e]1 orbit; group size=864;Yes
108216sg-216-161[ 4e4729 1bde4 28e1176]1 orbit; group size=432;Yes
108216sg-216-169[7d05ace 6366b39 19a4d2c]1 orbit; group size=864;Yes
108216sg-216-21[9ddb099 70c86ac 513f7ef]1 orbit; group size=1.9455550390e18;Yes
108216sg-216-88[8e89045 85f78db bd1d86a]1 orbit; group size=432;Yes
110220sg-220-10[9fac115 4a5e3ab 3ef0a34]1 orbit; group size=440;Yes
112224sg-224-12
sg-224-22
sg-224-23
sg-224-31
sg-224-37
[ 790bf2 36a7ac2 3a168f7]1 orbit; group size=8.0704505322e18;Yes
112224sg-224-40[eebca06 faade9f 209a25a]1 orbit; group size=1.5032385536e10;Yes
116232sg-232-12[af45e4b 4ded60c 5978b11]1 orbit; group size=464;Yes
120240sg-240-107[1ca2dfd 6ed5282 be5b556]2 orbits; group size=1.0555311627e15;Yes
120240sg-240-120[ddbc77f bbec82c 2942cdb]1 orbit; group size=480;Yes
120240sg-240-121[f2e6590 9371920 1945e9d]1 orbit; group size=480;Yes
120240sg-240-124[6ec3d01 ce8971e 85e24f9]1 orbit; group size=480;Yes
120240sg-240-189[cc74f72 bb31549 d423b3e]1 orbit; group size=960;Yes
120240sg-240-192[86a0b11 1909151 72309c9]2 orbits; group size=1440;Yes
120240sg-240-72
sg-240-73
sg-240-80
sg-240-197
[6b64c1a ca19b18 621d160]1 orbit; group size=1.3835058055e20;Yes
120240sg-240-91[5420c2d 59db06b c537d31]2 orbits; group size=2.5769803776e11;Yes
120240sg-240-95
sg-240-97
[b1a707b add6b8b 8582416]2 orbits; group size=7.7309411328e11;Yes
122244sg-244-3[25da4db e1d815 e9554bc]1 orbit; group size=488;Yes
126252sg-252-32[b1b098d 49616cb e245bf5]1 orbit; group size=1008;Yes