direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3⋊S3×C2×C12, C62.153D6, C12⋊7(S3×C6), C6⋊2(S3×C12), (C6×C12)⋊17C6, (C6×C12)⋊16S3, (C3×C12)⋊24D6, C62.76(C2×C6), C33⋊17(C22×C4), C32⋊9(C22×C12), (C32×C12)⋊17C22, (C3×C62).62C22, (C32×C6).86C23, C3⋊3(S3×C2×C12), (C3×C6×C12)⋊12C2, C6.53(S3×C2×C6), (C3×C6)⋊10(C4×S3), (C2×C12)⋊5(C3×S3), (C3×C6)⋊8(C2×C12), C32⋊17(S3×C2×C4), (C3×C12)⋊14(C2×C6), (C2×C6).76(S3×C6), C22.9(C6×C3⋊S3), C3⋊Dic3⋊12(C2×C6), (C2×C3⋊Dic3)⋊15C6, (C6×C3⋊Dic3)⋊19C2, (C32×C6)⋊11(C2×C4), (C22×C3⋊S3).9C6, C6.53(C22×C3⋊S3), (C6×C3⋊S3).66C22, (C3×C6).60(C22×C6), (C3×C6).175(C22×S3), (C3×C3⋊Dic3)⋊26C22, C2.1(C2×C6×C3⋊S3), (C2×C6×C3⋊S3).12C2, (C2×C6).68(C2×C3⋊S3), (C2×C3⋊S3).26(C2×C6), SmallGroup(432,711)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C3⋊S3×C2×C12 |
Generators and relations for C3⋊S3×C2×C12
G = < a,b,c,d,e | a2=b12=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 1060 in 388 conjugacy classes, 134 normal (22 characteristic)
C1, C2, C2, C2, C3, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C23, C32, C32, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C22×C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C2×C12, C2×C12, C22×S3, C22×C6, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C62, C62, C62, S3×C2×C4, C22×C12, C3×C3⋊S3, C32×C6, C32×C6, S3×C12, C6×Dic3, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C6×C12, C6×C12, S3×C2×C6, C22×C3⋊S3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, C3×C62, S3×C2×C12, C2×C4×C3⋊S3, C12×C3⋊S3, C6×C3⋊Dic3, C3×C6×C12, C2×C6×C3⋊S3, C3⋊S3×C2×C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, C12, D6, C2×C6, C22×C4, C3×S3, C3⋊S3, C4×S3, C2×C12, C22×S3, C22×C6, S3×C6, C2×C3⋊S3, S3×C2×C4, C22×C12, C3×C3⋊S3, S3×C12, C4×C3⋊S3, S3×C2×C6, C22×C3⋊S3, C6×C3⋊S3, S3×C2×C12, C2×C4×C3⋊S3, C12×C3⋊S3, C2×C6×C3⋊S3, C3⋊S3×C2×C12
►On 144 points
Generators in S144
(1 88)(2 89)(3 90)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 85)(11 86)(12 87)(13 105)(14 106)(15 107)(16 108)(17 97)(18 98)(19 99)(20 100)(21 101)(22 102)(23 103)(24 104)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 61)(37 127)(38 128)(39 129)(40 130)(41 131)(42 132)(43 121)(44 122)(45 123)(46 124)(47 125)(48 126)(49 112)(50 113)(51 114)(52 115)(53 116)(54 117)(55 118)(56 119)(57 120)(58 109)(59 110)(60 111)(73 136)(74 137)(75 138)(76 139)(77 140)(78 141)(79 142)(80 143)(81 144)(82 133)(83 134)(84 135)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132)(133 134 135 136 137 138 139 140 141 142 143 144)
(1 121 62)(2 122 63)(3 123 64)(4 124 65)(5 125 66)(6 126 67)(7 127 68)(8 128 69)(9 129 70)(10 130 71)(11 131 72)(12 132 61)(13 53 78)(14 54 79)(15 55 80)(16 56 81)(17 57 82)(18 58 83)(19 59 84)(20 60 73)(21 49 74)(22 50 75)(23 51 76)(24 52 77)(25 88 43)(26 89 44)(27 90 45)(28 91 46)(29 92 47)(30 93 48)(31 94 37)(32 95 38)(33 96 39)(34 85 40)(35 86 41)(36 87 42)(97 120 133)(98 109 134)(99 110 135)(100 111 136)(101 112 137)(102 113 138)(103 114 139)(104 115 140)(105 116 141)(106 117 142)(107 118 143)(108 119 144)
(1 66 129)(2 67 130)(3 68 131)(4 69 132)(5 70 121)(6 71 122)(7 72 123)(8 61 124)(9 62 125)(10 63 126)(11 64 127)(12 65 128)(13 74 57)(14 75 58)(15 76 59)(16 77 60)(17 78 49)(18 79 50)(19 80 51)(20 81 52)(21 82 53)(22 83 54)(23 84 55)(24 73 56)(25 47 96)(26 48 85)(27 37 86)(28 38 87)(29 39 88)(30 40 89)(31 41 90)(32 42 91)(33 43 92)(34 44 93)(35 45 94)(36 46 95)(97 141 112)(98 142 113)(99 143 114)(100 144 115)(101 133 116)(102 134 117)(103 135 118)(104 136 119)(105 137 120)(106 138 109)(107 139 110)(108 140 111)
(1 57)(2 58)(3 59)(4 60)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 129)(14 130)(15 131)(16 132)(17 121)(18 122)(19 123)(20 124)(21 125)(22 126)(23 127)(24 128)(25 133)(26 134)(27 135)(28 136)(29 137)(30 138)(31 139)(32 140)(33 141)(34 142)(35 143)(36 144)(37 103)(38 104)(39 105)(40 106)(41 107)(42 108)(43 97)(44 98)(45 99)(46 100)(47 101)(48 102)(61 81)(62 82)(63 83)(64 84)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(71 79)(72 80)(85 117)(86 118)(87 119)(88 120)(89 109)(90 110)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)
G:=sub<Sym(144)| (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,85)(11,86)(12,87)(13,105)(14,106)(15,107)(16,108)(17,97)(18,98)(19,99)(20,100)(21,101)(22,102)(23,103)(24,104)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,61)(37,127)(38,128)(39,129)(40,130)(41,131)(42,132)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,120)(58,109)(59,110)(60,111)(73,136)(74,137)(75,138)(76,139)(77,140)(78,141)(79,142)(80,143)(81,144)(82,133)(83,134)(84,135), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144), (1,121,62)(2,122,63)(3,123,64)(4,124,65)(5,125,66)(6,126,67)(7,127,68)(8,128,69)(9,129,70)(10,130,71)(11,131,72)(12,132,61)(13,53,78)(14,54,79)(15,55,80)(16,56,81)(17,57,82)(18,58,83)(19,59,84)(20,60,73)(21,49,74)(22,50,75)(23,51,76)(24,52,77)(25,88,43)(26,89,44)(27,90,45)(28,91,46)(29,92,47)(30,93,48)(31,94,37)(32,95,38)(33,96,39)(34,85,40)(35,86,41)(36,87,42)(97,120,133)(98,109,134)(99,110,135)(100,111,136)(101,112,137)(102,113,138)(103,114,139)(104,115,140)(105,116,141)(106,117,142)(107,118,143)(108,119,144), (1,66,129)(2,67,130)(3,68,131)(4,69,132)(5,70,121)(6,71,122)(7,72,123)(8,61,124)(9,62,125)(10,63,126)(11,64,127)(12,65,128)(13,74,57)(14,75,58)(15,76,59)(16,77,60)(17,78,49)(18,79,50)(19,80,51)(20,81,52)(21,82,53)(22,83,54)(23,84,55)(24,73,56)(25,47,96)(26,48,85)(27,37,86)(28,38,87)(29,39,88)(30,40,89)(31,41,90)(32,42,91)(33,43,92)(34,44,93)(35,45,94)(36,46,95)(97,141,112)(98,142,113)(99,143,114)(100,144,115)(101,133,116)(102,134,117)(103,135,118)(104,136,119)(105,137,120)(106,138,109)(107,139,110)(108,140,111), (1,57)(2,58)(3,59)(4,60)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,129)(14,130)(15,131)(16,132)(17,121)(18,122)(19,123)(20,124)(21,125)(22,126)(23,127)(24,128)(25,133)(26,134)(27,135)(28,136)(29,137)(30,138)(31,139)(32,140)(33,141)(34,142)(35,143)(36,144)(37,103)(38,104)(39,105)(40,106)(41,107)(42,108)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(61,81)(62,82)(63,83)(64,84)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80)(85,117)(86,118)(87,119)(88,120)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)>;
G:=Group( (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,85)(11,86)(12,87)(13,105)(14,106)(15,107)(16,108)(17,97)(18,98)(19,99)(20,100)(21,101)(22,102)(23,103)(24,104)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,61)(37,127)(38,128)(39,129)(40,130)(41,131)(42,132)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,120)(58,109)(59,110)(60,111)(73,136)(74,137)(75,138)(76,139)(77,140)(78,141)(79,142)(80,143)(81,144)(82,133)(83,134)(84,135), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144), (1,121,62)(2,122,63)(3,123,64)(4,124,65)(5,125,66)(6,126,67)(7,127,68)(8,128,69)(9,129,70)(10,130,71)(11,131,72)(12,132,61)(13,53,78)(14,54,79)(15,55,80)(16,56,81)(17,57,82)(18,58,83)(19,59,84)(20,60,73)(21,49,74)(22,50,75)(23,51,76)(24,52,77)(25,88,43)(26,89,44)(27,90,45)(28,91,46)(29,92,47)(30,93,48)(31,94,37)(32,95,38)(33,96,39)(34,85,40)(35,86,41)(36,87,42)(97,120,133)(98,109,134)(99,110,135)(100,111,136)(101,112,137)(102,113,138)(103,114,139)(104,115,140)(105,116,141)(106,117,142)(107,118,143)(108,119,144), (1,66,129)(2,67,130)(3,68,131)(4,69,132)(5,70,121)(6,71,122)(7,72,123)(8,61,124)(9,62,125)(10,63,126)(11,64,127)(12,65,128)(13,74,57)(14,75,58)(15,76,59)(16,77,60)(17,78,49)(18,79,50)(19,80,51)(20,81,52)(21,82,53)(22,83,54)(23,84,55)(24,73,56)(25,47,96)(26,48,85)(27,37,86)(28,38,87)(29,39,88)(30,40,89)(31,41,90)(32,42,91)(33,43,92)(34,44,93)(35,45,94)(36,46,95)(97,141,112)(98,142,113)(99,143,114)(100,144,115)(101,133,116)(102,134,117)(103,135,118)(104,136,119)(105,137,120)(106,138,109)(107,139,110)(108,140,111), (1,57)(2,58)(3,59)(4,60)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,129)(14,130)(15,131)(16,132)(17,121)(18,122)(19,123)(20,124)(21,125)(22,126)(23,127)(24,128)(25,133)(26,134)(27,135)(28,136)(29,137)(30,138)(31,139)(32,140)(33,141)(34,142)(35,143)(36,144)(37,103)(38,104)(39,105)(40,106)(41,107)(42,108)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(61,81)(62,82)(63,83)(64,84)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80)(85,117)(86,118)(87,119)(88,120)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116) );
G=PermutationGroup([[(1,88),(2,89),(3,90),(4,91),(5,92),(6,93),(7,94),(8,95),(9,96),(10,85),(11,86),(12,87),(13,105),(14,106),(15,107),(16,108),(17,97),(18,98),(19,99),(20,100),(21,101),(22,102),(23,103),(24,104),(25,62),(26,63),(27,64),(28,65),(29,66),(30,67),(31,68),(32,69),(33,70),(34,71),(35,72),(36,61),(37,127),(38,128),(39,129),(40,130),(41,131),(42,132),(43,121),(44,122),(45,123),(46,124),(47,125),(48,126),(49,112),(50,113),(51,114),(52,115),(53,116),(54,117),(55,118),(56,119),(57,120),(58,109),(59,110),(60,111),(73,136),(74,137),(75,138),(76,139),(77,140),(78,141),(79,142),(80,143),(81,144),(82,133),(83,134),(84,135)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132),(133,134,135,136,137,138,139,140,141,142,143,144)], [(1,121,62),(2,122,63),(3,123,64),(4,124,65),(5,125,66),(6,126,67),(7,127,68),(8,128,69),(9,129,70),(10,130,71),(11,131,72),(12,132,61),(13,53,78),(14,54,79),(15,55,80),(16,56,81),(17,57,82),(18,58,83),(19,59,84),(20,60,73),(21,49,74),(22,50,75),(23,51,76),(24,52,77),(25,88,43),(26,89,44),(27,90,45),(28,91,46),(29,92,47),(30,93,48),(31,94,37),(32,95,38),(33,96,39),(34,85,40),(35,86,41),(36,87,42),(97,120,133),(98,109,134),(99,110,135),(100,111,136),(101,112,137),(102,113,138),(103,114,139),(104,115,140),(105,116,141),(106,117,142),(107,118,143),(108,119,144)], [(1,66,129),(2,67,130),(3,68,131),(4,69,132),(5,70,121),(6,71,122),(7,72,123),(8,61,124),(9,62,125),(10,63,126),(11,64,127),(12,65,128),(13,74,57),(14,75,58),(15,76,59),(16,77,60),(17,78,49),(18,79,50),(19,80,51),(20,81,52),(21,82,53),(22,83,54),(23,84,55),(24,73,56),(25,47,96),(26,48,85),(27,37,86),(28,38,87),(29,39,88),(30,40,89),(31,41,90),(32,42,91),(33,43,92),(34,44,93),(35,45,94),(36,46,95),(97,141,112),(98,142,113),(99,143,114),(100,144,115),(101,133,116),(102,134,117),(103,135,118),(104,136,119),(105,137,120),(106,138,109),(107,139,110),(108,140,111)], [(1,57),(2,58),(3,59),(4,60),(5,49),(6,50),(7,51),(8,52),(9,53),(10,54),(11,55),(12,56),(13,129),(14,130),(15,131),(16,132),(17,121),(18,122),(19,123),(20,124),(21,125),(22,126),(23,127),(24,128),(25,133),(26,134),(27,135),(28,136),(29,137),(30,138),(31,139),(32,140),(33,141),(34,142),(35,143),(36,144),(37,103),(38,104),(39,105),(40,106),(41,107),(42,108),(43,97),(44,98),(45,99),(46,100),(47,101),(48,102),(61,81),(62,82),(63,83),(64,84),(65,73),(66,74),(67,75),(68,76),(69,77),(70,78),(71,79),(72,80),(85,117),(86,118),(87,119),(88,120),(89,109),(90,110),(91,111),(92,112),(93,113),(94,114),(95,115),(96,116)]])
144 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | ··· | 3N | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | ··· | 6AP | 6AQ | ··· | 6AX | 12A | ··· | 12H | 12I | ··· | 12BD | 12BE | ··· | 12BL |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C12 | S3 | D6 | D6 | C3×S3 | C4×S3 | S3×C6 | S3×C6 | S3×C12 |
| kernel | C3⋊S3×C2×C12 | C12×C3⋊S3 | C6×C3⋊Dic3 | C3×C6×C12 | C2×C6×C3⋊S3 | C2×C4×C3⋊S3 | C6×C3⋊S3 | C4×C3⋊S3 | C2×C3⋊Dic3 | C6×C12 | C22×C3⋊S3 | C2×C3⋊S3 | C6×C12 | C3×C12 | C62 | C2×C12 | C3×C6 | C12 | C2×C6 | C6 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 8 | 8 | 2 | 2 | 2 | 16 | 4 | 8 | 4 | 8 | 16 | 16 | 8 | 32 |
Matrix representation of C3⋊S3×C2×C12 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 11 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 4 | 3 |
| 3 | 3 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 2 | 12 | 0 | 0 |
| 0 | 0 | 1 | 5 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[11,0,0,0,0,11,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,9,4,0,0,0,3],[3,0,0,0,3,9,0,0,0,0,1,0,0,0,0,1],[1,2,0,0,0,12,0,0,0,0,1,0,0,0,5,12] >;
C3⋊S3×C2×C12 in GAP, Magma, Sage, TeX
C_3\rtimes S_3\times C_2\times C_{12} % in TeX
G:=Group("C3:S3xC2xC12"); // GroupNames label
G:=SmallGroup(432,711);
// by ID
G=gap.SmallGroup(432,711);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,142,4037,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^12=c^3=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations