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 32 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 |
| 2 | 5 | sg-5-1 | [ 44 9a902 248b4d] | 1 orbit; group size=2; | No |
| 10 | 25 | sg-25-2 | [ 17ec 6f7beba 74b941b] | 1 orbit; group size=28800; | Yes |
| 22 | 55 | sg-55-1 | [84ca24d e37024c b3ceb4c] | 1 orbit; group size=1320; | Yes |
| 24 | 60 | sg-60-5 | [a682f88 fe69114 96269a9] | 1 orbit; group size=480; | Yes |
| 32 | 80 | sg-80-49 | [e632901 3fdc154 caf3957] | 1 orbit; group size=3840; | Yes |
| 48 | 120 | sg-120-5 | [cec8526 dc9b56e 618613b] | 1 orbit; group size=960; | Yes |
| 50 | 125 | sg-125-3 | [8c13162 f781a4 fc6d418] | 1 orbit; group size=4000; | Yes |
| 62 | 155 | sg-155-1 | [973e140 27e0787 1e325ea] | 1 orbit; group size=310; | Yes |
| 64 | 160 | sg-160-199 | [9cf125c 55ae425 1b29490] | 1 orbit; group size=640; | Yes |
| 82 | 205 | sg-205-1 | [239a563 d5ab3d2 e0c2c7] | 1 orbit; group size=410; | Yes |
| 110 | 275 | sg-275-3 | [e02bd46 52ad388 eca7d67] | 1 orbit; group size=550; | Yes |
| 120 | 300 | sg-300-22 | [105963d c545b09 6297140] | 2 orbits; group size=1200; | Yes |
| 122 | 305 | sg-305-1 | [654d2a5 746b57e fef3dc] | 1 orbit; group size=610; | Yes |
| 128 | 320 | sg-320-1012 | [5b012c9 6160bc8 f44a2cc] | 1 orbit; group size=2560; | Yes |
| 142 | 355 | sg-355-1 | [db90525 6d0d695 84c9a72] | 1 orbit; group size=710; | Yes |
| 144 | 360 | sg-360-118 | [a6c3285 b470740 d9b5ee0] | 1 orbit; group size=5760; | Yes |
| 160 | 400 | sg-400-213 | [cdd3ce8 b85abb0 d822e2] | 1 orbit; group size=3200; | Yes |
| 162 | 405 | sg-405-15 | [78bd51f beaf7bc 1dfc116] | 1 orbit; group size=19440; | Yes |
| 202 | 505 | sg-505-1 | [b84d3e9 16def22 49d50c] | 1 orbit; group size=1010; | Yes |
| 240 | 600 | sg-600-54 | [1cc14af 63c66b5 e3b98f] | 2 orbits; group size=2400; | Yes |
| 242 | 605 | sg-605-1 | [b654e37 ec9390a 889ea2f] | 1 orbit; group size=1210; | Yes |
| 242 | 605 | sg-605-5 | [86821dc 30e35ea 93922ba] | 1 orbit; group size=1210; | Yes |
| 242 | 605 | sg-605-6 | [b25419e 844d2ef 7045aba] | 1 orbit; group size=2420; | Yes |
| 250 | 625 | sg-625-7 | [faf53ab 2951997 5181f7b] | 2 orbits; group size=10000; | Yes |
| 262 | 655 | sg-655-1 | [ f57274 da877ae bc0b31c] | 1 orbit; group size=1310; | Yes |
| 264 | 660 | sg-660-13 | [7a4e488 95005d8 215b670] | 1 orbit; group size=5280; | Yes |
| 288 | 720 | sg-720-409 | [ad4452e 7bb48c8 308676b] | 1 orbit; group size=2880; | Yes |
| 302 | 755 | sg-755-1 | [ebc6451 657b8fa a1a7211] | 1 orbit; group size=1510; | Yes |
| 310 | 775 | sg-775-3 | [dd864a2 4f240f1 375e5ac] | 1 orbit; group size=1550; | Yes |
| 320 | 800 | sg-800-1065 | [7203dd4 7ac19ff d517839] | 1 orbit; group size=6400; | Yes |
| 352 | 880 | sg-880-214 | [554549c b77c94b 86321b4] | 1 orbit; group size=1760; | Yes |
| 362 | 905 | sg-905-1 | [5ffd5ea 44a5ca1 d96e357] | 1 orbit; group size=1810; | Yes |
| 382 | 955 | sg-955-1 | [c7ca940 a476869 19c3807] | 1 orbit; group size=1910; | Yes |
| 384 | 960 | sg-960-11357 | [b731617 7f11e73 de30654] | 1 orbit; group size=7680; | Yes |
| 384 | 960 | sg-960-11358 | [a59b657 4d0c38 ba578e0] | 1 orbit; group size=46080; | Yes |
| 410 | 1025 | sg-1025-3 | [ da0ba0 f0683f dab745b] | 1 orbit; group size=2050; | Yes |
| 422 | 1055 | sg-1055-1 | [905c63a 728fdd4 392ac38] | 1 orbit; group size=2110; | Yes |
| 432 | 1080 | sg-1080-260 | [8471367 65d4d11 8d34325] | 1 orbit; group size=4320; | Yes |
| 482 | 1205 | sg-1205-1 | [ 9a5503 1462a9e 9ca8fa8] | 1 orbit; group size=2410; | Yes |
| 486 | 1215 | sg-1215-68 | [87973f9 2107d3d 673a23d] | 1 orbit; group size=4860; | Yes |
| 502 | 1255 | sg-1255-1 | [16ca8d6 cb8f523 37c1316] | 1 orbit; group size=2510; | Yes |
| 528 | 1320 | sg-1320-13 | [29cac7d 2cd202a 579582d] | 1 orbit; group size=5280; | Yes |
| 542 | 1355 | sg-1355-1 | [37ef318 36a5e41 28e017] | 1 orbit; group size=2710; | Yes |
| 550 | 1375 | sg-1375-9 | [969847d f6a526f d21a28c] | 1 orbit; group size=2750; | Yes |
| 562 | 1405 | sg-1405-1 | [57e48cd 745c663 5297ed2] | 1 orbit; group size=2810; | Yes |
| 600 | 1500 | sg-1500-112 | [bf23345 abae174 c35723f] | 1 orbit; group size=12000; | Yes |
| 610 | 1525 | sg-1525-3 | [916739a ddedb6e f2150bf] | 1 orbit; group size=3050; | Yes |
| 622 | 1555 | sg-1555-1 | [5a282c9 f6f8170 f93d10e] | 1 orbit; group size=3110 | Yes |
| 640 | 1600 | sg-1600-6786 | [e6a21de 94d5614 a11e55e] | 1 orbit; group size=12800; | Yes |
| 662 | 1655 | sg-1655-1 | [55f5a6d d763c1b 356199e] | 1 orbit; group size=3310; | Yes |
| 682 | 1705 | sg-1705-3 | [c234669 ef4d649 b19b2b2] | 1 orbit; group size=3410; | Yes |
| 682 | 1705 | sg-1705-4 | [72cd144 b7cf15e 405ce60] | 1 orbit; group size=3410; | Yes |
| 682 | 1705 | sg-1705-5 | [f1ce571 452cc29 31fbcc3] | 1 orbit; group size=3410; | Yes |
| 682 | 1705 | sg-1705-6 | [66328f7 5a926a4 3e3edcd] | 1 orbit; group size=3410; | Yes |
| 704 | 1760 | sg-1760-1139 | [5098afa 21391a0 aaea452] | 1 orbit; group size=3520; | Yes |
| 710 | 1775 | sg-1775-3 | [abadb68 e2a501 ce90797] | 1 orbit; group size=3550; | Yes |
| 720 | 1800 | sg-1800-555 | [a983e28 57fbbfe e198e0f] | 2 orbits; group size=14400; | Yes |
| 722 | 1805 | sg-1805-2 | [60b3990 7757b91 aba498d] | 1 orbit; group size=7220; | Yes |
| 800 | 2000 | sg-2000-488 | [2c52f03 48caf84 cd54c3b] | 1 orbit; group size=16000; | Yes |