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) | G | Nauty Key | Automorphism group | Simple |
| 8 | 16 | sg-16-2 sg-16-3 sg-16-4 | [ 1bec 36a0a8e d566c4c] | 1 orbit; group size=1152; | Yes |
| 10 | 20 | sg-20-3 | [ 197ec acba481 527bf19] | 1 orbit; group size=240; | Yes |
| 12 | 24 | sg-24-12 | [ 107ec 3785a74 4cbbaed] | 1 orbit; group size=768; | Yes |
| 16 | 32 | sg-32-2 sg-32-10 sg-32-11 sg-32-13 | [ 2389ec 761d8d3 9b71d9a] | 1 orbit; group size=4096; | Yes |
| 16 | 32 | sg-32-6 | [ 2bb8ec 23a2f94 4eefea5] | 1 orbit; group size=384; | Yes |
| 18 | 36 | sg-36-9 | [fdc1acb ea9fdcb 70f0b7a] | 1 orbit; group size=144; | Yes |
| 20 | 40 | sg-40-12 | [b12e2e0 8cbf445 7d9f1d4] | 1 orbit; group size=80; | Yes |
| 24 | 48 | sg-48-11 sg-48-12 sg-48-19 sg-48-48 | [e18db91 b870c97 a8b037f] | 1 orbit; group size=98304; | Yes |
| 24 | 48 | sg-48-30 | [33a2afc ffb72ec ab7d92d] | 2 orbits; group size=3072; | Yes |
| 26 | 52 | sg-52-3 | [b27d6b9 4981e0f ff23b0f] | 1 orbit; group size=104; | Yes |
| 30 | 60 | sg-60-7 | [ 8dcddd a885ace ba0156d] | 1 orbit; group size=720; | Yes |
| 32 | 64 | sg-64-18 | [865a651 8f844e4 f22c737] | 1 orbit; group size=4096; | Yes |
| 32 | 64 | sg-64-34 | [e0299e1 22ece48 88cef4] | 1 orbit; group size=256; | Yes |
| 32 | 64 | sg-64-8 sg-64-21 sg-64-39 sg-64-41 sg-64-48 | [39a0d28 34b79b2 28b5f1b] | 1 orbit; group size=2097152; | Yes |
| 32 | 64 | sg-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 |
| 34 | 68 | sg-68-3 | [b8319c9 1c8ecd4 703be37] | 1 orbit; group size=136; | Yes |
| 36 | 72 | sg-72-15 | [b1e7130 7f5af6b 9c6aaca] | 1 orbit; group size=9437184; | Yes |
| 36 | 72 | sg-72-41 | [eb4a1c6 b1e575d f4d071b] | 1 orbit; group size=144; | Yes |
| 36 | 72 | sg-72-45 | [626f92a a310a6c 20caa1a] | 1 orbit; group size=288; | Yes |
| 40 | 80 | sg-80-11 sg-80-12 sg-80-19 | [ 29ce33 bf2fc2e a04236a] | 1 orbit; group size=41943040; | Yes |
| 40 | 80 | sg-80-30 | [abd426c 801afeb 70a9c2f] | 1 orbit; group size=160; | Yes |
| 40 | 80 | sg-80-31 | [4208c34 225ec90 3d0f4b7] | 1 orbit; group size=160; | Yes |
| 40 | 80 | sg-80-34 | [e6bab40 3d403e8 9168d21] | 1 orbit; group size=160; | Yes |
| 48 | 96 | sg-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 |
| 48 | 96 | sg-96-185 sg-96-186 sg-96-194 sg-96-195 | [ddfde41 25422e1 76634f5] | 2 orbits; group size=3145728; | Yes |
| 48 | 96 | sg-96-227 | [7ad40f3 f20f82a 481ce5] | 1 orbit; group size=768; | Yes |
| 48 | 96 | sg-96-41 | [24f9da7 6d7e3d6 2a795f1] | 1 orbit; group size=98304; | Yes |
| 48 | 96 | sg-96-67 | [7cfeeaf 764531b 5e79462] | 1 orbit; group size=384; | Yes |
| 50 | 100 | sg-100-10 | [a9a44c5 6c9d569 3178280] | 1 orbit; group size=200; | Yes |
| 50 | 100 | sg-100-12 | [1a3d9e1 233229c 6db4d3a] | 1 orbit; group size=400; | Yes |
| 50 | 100 | sg-100-3 | [4763b37 9c095d 4da592c] | 1 orbit; group size=200; | Yes |
| 52 | 104 | sg-104-12 | [9a33b70 9c7132f ad6e409] | 1 orbit; group size=208; | Yes |
| 54 | 108 | sg-108-15 | [c3f8fab c8e6f9d a11b82d] | 1 orbit; group size=432; | Yes |
| 54 | 108 | sg-108-37 | [9e80297 9746eee 2920284] | 1 orbit; group size=1296; | Yes |
| 56 | 112 | sg-112-10 sg-112-11 sg-112-18 | [8d3a57f b88fb1f 5c52d54] | 1 orbit; group size=1.5032385536e10; | Yes |
| 58 | 116 | sg-116-3 | [aec78cc 241d052 df8460c] | 1 orbit; group size=232; | Yes |
| 60 | 120 | sg-120-34 | [5f68fa2 44bf120 4daaf34] | 1 orbit; group size=480; | Yes |
| 60 | 120 | sg-120-38 | [ 420015 177d68 2d5d8bb] | 1 orbit; group size=6.4424509440e10; | Yes |
| 60 | 120 | sg-120-41 | [a53840c 9bf0f60 c561c97] | 1 orbit; group size=240; | Yes |
| 64 | 128 | sg-128-122 sg-128-134 sg-128-136 sg-128-139 sg-128-141 | [720e9a0 8d596e8 edbc284] | 2 orbits; group size=25165824; | Yes |
| 64 | 128 | sg-128-123 | [2047b22 6452816 57caf32] | 1 orbit; group size=512; | Yes |
| 64 | 128 | sg-128-140 | [6de82ec 372ac90 1489f84] | 1 orbit; group size=512; | Yes |
| 64 | 128 | sg-128-144 sg-128-146 | [db22e7d ad3904b 47b7da4] | 2 orbits; group size=2048; | Yes |
| 64 | 128 | sg-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 |
| 64 | 128 | sg-128-21 sg-128-116 | [5c22c75 5b6784 df5f16c] | 1 orbit; group size=2097152; 1 10 | Yes |
| 64 | 128 | sg-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 |
| 64 | 128 | sg-128-79 sg-128-112 sg-128-148 sg-128-150 sg-128-156 | [2aeeb58 8a79685 9109172] | 1 orbit; group size=2.7487790694e11; | Yes |
| 68 | 136 | sg-136-13 | [f7d83ca c58807b 22a8f4a] | 1 orbit; group size=272; | Yes |
| 70 | 140 | sg-140-6 | [ 145b91 116c5a8 26db4] | 1 orbit; group size=280; | Yes |
| 72 | 144 | sg-144-11 sg-144-12 sg-144-19 sg-144-109 | [ 5347b0 7ba3880 dd4b03c] | 1 orbit; group size=4.9478023250e12; | Yes |
| 72 | 144 | sg-144-115 sg-144-116 | [4459ecf f4e0c54 f6ded4f] | 2 orbits; group size=37748736; | Yes |
| 72 | 144 | sg-144-120 | [3034fca 6a2d429 b9025da] | 2 orbits; group size=144; | Yes |
| 72 | 144 | sg-144-132 | [196cd50 81d9c1f f227174] | 1 orbit; group size=576; | Yes |
| 72 | 144 | sg-144-133 | [5e1b0c7 9bdb63d db0147e] | 1 orbit; group size=576; | Yes |
| 72 | 144 | sg-144-136 | [e6ccc97 56302fe e7cb151] | 1 orbit; group size=576; | Yes |
| 72 | 144 | sg-144-33 | [137deb7 e3ee03d 44836e2] | 2 orbits; group size=2415919104; | Yes |
| 74 | 148 | sg-148-3 | [36f79dd e1e2bb2 8efe195] | 1 orbit; group size=296; | Yes |
| 78 | 156 | sg-156-10 | [ba5d000 5f1996b fdbb9c9] | 1 orbit; group size=312; | Yes |
| 80 | 160 | sg-160-13 sg-160-23 sg-160-24 sg-160-32 sg-160-38 | [8185ebb fabe6a3 ddefee2] | 1 orbit; group size=8.7960930222e13; | Yes |
| 80 | 160 | sg-160-234 | [9528797 4c6c3ca 12a97d4] | 2 orbits; group size=335544320; | Yes |
| 80 | 160 | sg-160-41 | [c039596 1e41f50 6416eb2] | 1 orbit; group size=41943040; | Yes |
| 80 | 160 | sg-160-68 | [7002bdd f63dac3 e1a99c] | 1 orbit; group size=320; | Yes |
| 80 | 160 | sg-160-69 | [7be32d6 1caff41 6add779] | 1 orbit; group size=320; | Yes |
| 80 | 160 | sg-160-74 | [732ac42 2c47d25 713ba7a] | 1 orbit; group size=320; | Yes |
| 80 | 160 | sg-160-81 | [961719e 1ad3de0 f5a3629] | 1 orbit; group size=320; | Yes |
| 80 | 160 | sg-160-83 sg-160-84 | [6221b7a 61d0e8c 244a101] | 2 orbits; group size=83886080; | Yes |
| 80 | 160 | sg-160-86 | [c54bc9f 466f8fa 8b4047] | 1 orbit; group size=320; | Yes |
| 82 | 164 | sg-164-3 | [3e9e636 e2a19f8 bf04140] | 1 orbit; group size=328; | Yes |
| 84 | 168 | sg-168-42 | [77a6a75 b27c238 cbf2152] | 1 orbit; group size=672; | Yes |
| 84 | 168 | sg-168-46 | [9b428eb 2efd8fa 9842b07] | 1 orbit; group size=3.6943590693e14; | Yes |
| 88 | 176 | sg-176-10 sg-176-11 sg-176-18 | [dba8cfb ed026d d9046b0] | 1 orbit; group size=1.5481123719e15; | Yes |
| 90 | 180 | sg-180-24 | [cfc0da3 688ead9 1d5edd8] | 1 orbit; group size=720; | Yes |
| 90 | 180 | sg-180-25 | [e8c7524 696f9d5 fd3ddec] | 1 orbit; group size=360; | Yes |
| 90 | 180 | sg-180-6 | [f9543d8 9954efe 4295e70] | 1 orbit; group size=360; | Yes |
| 96 | 192 | sg-192-100 | [ba4a49c 942519b c0fd5df] | 1 orbit; group size=768; | Yes |
| 96 | 192 | sg-192-104 | [87d055a 19f5348 584884] | 1 orbit; group size=768; | Yes |
| 96 | 192 | sg-192-1491 | [417d41c 386268b ede2a2a] | 1 orbit; group size=1536; | Yes |
| 96 | 192 | sg-192-1493 | [2bf2242 e18a4d2 38b5fd0] | 1 orbit; group size=768; | Yes |
| 96 | 192 | sg-192-1495 | [6cc3de2 73c7799 c0e79b0] | 1 orbit; group size=768; | Yes |
| 96 | 192 | sg-192-1538 | [ededeb5 d5895e7 d45e489] | 1 orbit; group size=98304; | Yes |
| 96 | 192 | sg-192-185 | [3dabc11 c438fc9 93ec60f] | 2 orbits; group size=192; | Yes |
| 96 | 192 | sg-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 |
| 96 | 192 | sg-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 |
| 96 | 192 | sg-192-85 sg-192-116 | [5653297 8804ea9 a64d065] | 1 orbit; group size=805306368; | Yes |
| 96 | 192 | sg-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 |
| 96 | 192 | sg-192-981 | [100d0a2 a2ac7ee e2520c9] | 1 orbit; group size=768; | Yes |
| 96 | 192 | sg-192-982 | [f97d8f7 b9c3e0d 769a29c] | 1 orbit; group size=768; | Yes |
| 98 | 196 | sg-196-8 | [8f20346 cd3605e 493641a] | 1 orbit; group size=784; | Yes |
| 100 | 200 | sg-200-12 | [e843202 b21e109 5674947] | 1 orbit; group size=400; | Yes |
| 100 | 200 | sg-200-42 | [a61b969 f95de98 bdb3b73] | 2 orbits; group size=400; | Yes |
| 100 | 200 | sg-200-44 | [7dc484e 38e5966 b1c2060] | 1 orbit; group size=800; | Yes |
| 100 | 200 | sg-200-46 | [4ec49f8 91ecad7 5e45595] | 1 orbit; group size=400; | Yes |
| 100 | 200 | sg-200-48 | [10c8431 a1d7443 c0d0ca8] | 1 orbit; group size=800; | Yes |
| 102 | 204 | sg-204-6 | [c12ff57 d52ad23 d26b0b4] | 1 orbit; group size=408; | Yes |
| 104 | 208 | sg-208-11 sg-208-12 sg-208-19 | [b88c91c 6ea041b 2d1128a] | 1 orbit; group size=4.6837436125e17; | Yes |
| 104 | 208 | sg-208-30 | [c4a8bec 556762f 4dacfc3] | 1 orbit; group size=416; | Yes |
| 104 | 208 | sg-208-31 | [ fba0c0 d34eaba df840c2] | 1 orbit; group size=416; | Yes |
| 104 | 208 | sg-208-34 | [2fb929e 2343951 59661ec] | 1 orbit; group size=416; | Yes |
| 106 | 212 | sg-212-3 | [d391af6 8f4eb90 8558e97] | 1 orbit; group size=424; | Yes |
| 108 | 216 | sg-216-100 | [55bdafa 40a336b d3dbe9e] | 1 orbit; group size=864; | Yes |
| 108 | 216 | sg-216-161 | [ 4e4729 1bde4 28e1176] | 1 orbit; group size=432; | Yes |
| 108 | 216 | sg-216-169 | [7d05ace 6366b39 19a4d2c] | 1 orbit; group size=864; | Yes |
| 108 | 216 | sg-216-21 | [9ddb099 70c86ac 513f7ef] | 1 orbit; group size=1.9455550390e18; | Yes |
| 108 | 216 | sg-216-88 | [8e89045 85f78db bd1d86a] | 1 orbit; group size=432; | Yes |
| 110 | 220 | sg-220-10 | [9fac115 4a5e3ab 3ef0a34] | 1 orbit; group size=440; | Yes |
| 112 | 224 | sg-224-12 sg-224-22 sg-224-23 sg-224-31 sg-224-37 | [ 790bf2 36a7ac2 3a168f7] | 1 orbit; group size=8.0704505322e18; | Yes |
| 112 | 224 | sg-224-40 | [eebca06 faade9f 209a25a] | 1 orbit; group size=1.5032385536e10; | Yes |
| 116 | 232 | sg-232-12 | [af45e4b 4ded60c 5978b11] | 1 orbit; group size=464; | Yes |
| 120 | 240 | sg-240-107 | [1ca2dfd 6ed5282 be5b556] | 2 orbits; group size=1.0555311627e15; | Yes |
| 120 | 240 | sg-240-120 | [ddbc77f bbec82c 2942cdb] | 1 orbit; group size=480; | Yes |
| 120 | 240 | sg-240-121 | [f2e6590 9371920 1945e9d] | 1 orbit; group size=480; | Yes |
| 120 | 240 | sg-240-124 | [6ec3d01 ce8971e 85e24f9] | 1 orbit; group size=480; | Yes |
| 120 | 240 | sg-240-189 | [cc74f72 bb31549 d423b3e] | 1 orbit; group size=960; | Yes |
| 120 | 240 | sg-240-192 | [86a0b11 1909151 72309c9] | 2 orbits; group size=1440; | Yes |
| 120 | 240 | sg-240-72 sg-240-73 sg-240-80 sg-240-197 | [6b64c1a ca19b18 621d160] | 1 orbit; group size=1.3835058055e20; | Yes |
| 120 | 240 | sg-240-91 | [5420c2d 59db06b c537d31] | 2 orbits; group size=2.5769803776e11; | Yes |
| 120 | 240 | sg-240-95 sg-240-97 | [b1a707b add6b8b 8582416] | 2 orbits; group size=7.7309411328e11; | Yes |
| 122 | 244 | sg-244-3 | [25da4db e1d815 e9554bc] | 1 orbit; group size=488; | Yes |
| 126 | 252 | sg-252-32 | [b1b098d 49616cb e245bf5] | 1 orbit; group size=1008; | Yes |