direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4.Dic6, C6⋊1(C4.Q8), C4⋊C4.224D6, C12.15(C4⋊C4), C12.62(C2×Q8), (C2×C12).15Q8, (C2×C12).131D4, (C2×C4).27Dic6, C4.27(C2×Dic6), (C2×C6).39SD16, C6.65(C2×SD16), C12.57(C22×C4), (C22×C4).344D6, (C22×C6).181D4, C4.15(Dic3⋊C4), (C2×C12).316C23, C23.103(C3⋊D4), C4⋊Dic3.322C22, C22.10(D4.S3), C22.24(Dic3⋊C4), (C22×C12).131C22, C22.10(Q8⋊2S3), (C2×C3⋊C8)⋊6C4, C3⋊C8⋊24(C2×C4), C3⋊2(C2×C4.Q8), C4.85(S3×C2×C4), (C6×C4⋊C4).4C2, C6.33(C2×C4⋊C4), (C2×C4⋊C4).5S3, (C22×C3⋊C8).4C2, C2.1(C2×D4.S3), (C2×C6).45(C4⋊C4), (C2×C4).150(C4×S3), (C2×C12).75(C2×C4), (C2×C6).436(C2×D4), C2.8(C2×Dic3⋊C4), C2.1(C2×Q8⋊2S3), (C2×C3⋊C8).240C22, (C2×C4⋊Dic3).30C2, C22.55(C2×C3⋊D4), (C2×C4).123(C3⋊D4), (C3×C4⋊C4).255C22, (C2×C4).416(C22×S3), SmallGroup(192,522)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4.Dic6
G = < a,b,c,d | a2=b4=c12=1, d2=bc6, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c-1 >
Subgroups: 280 in 130 conjugacy classes, 79 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C6 [×3], C6 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×8], C23, Dic3 [×2], C12 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×6], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×6], C22×C4, C22×C4 [×2], C3⋊C8 [×4], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12 [×4], C22×C6, C4.Q8 [×4], C2×C4⋊C4, C2×C4⋊C4, C22×C8, C2×C3⋊C8 [×6], C4⋊Dic3 [×2], C4⋊Dic3, C3×C4⋊C4 [×2], C3×C4⋊C4, C22×Dic3, C22×C12, C22×C12, C2×C4.Q8, C4.Dic6 [×4], C22×C3⋊C8, C2×C4⋊Dic3, C6×C4⋊C4, C2×C4.Dic6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×3], C4⋊C4 [×4], SD16 [×4], C22×C4, C2×D4, C2×Q8, Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C4.Q8 [×4], C2×C4⋊C4, C2×SD16 [×2], Dic3⋊C4 [×4], D4.S3 [×2], Q8⋊2S3 [×2], C2×Dic6, S3×C2×C4, C2×C3⋊D4, C2×C4.Q8, C4.Dic6 [×4], C2×Dic3⋊C4, C2×D4.S3, C2×Q8⋊2S3, C2×C4.Dic6
►Regular action on 192 points
Generators in S192
(1 60)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 58)(12 59)(13 155)(14 156)(15 145)(16 146)(17 147)(18 148)(19 149)(20 150)(21 151)(22 152)(23 153)(24 154)(25 129)(26 130)(27 131)(28 132)(29 121)(30 122)(31 123)(32 124)(33 125)(34 126)(35 127)(36 128)(37 96)(38 85)(39 86)(40 87)(41 88)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(61 97)(62 98)(63 99)(64 100)(65 101)(66 102)(67 103)(68 104)(69 105)(70 106)(71 107)(72 108)(73 169)(74 170)(75 171)(76 172)(77 173)(78 174)(79 175)(80 176)(81 177)(82 178)(83 179)(84 180)(109 134)(110 135)(111 136)(112 137)(113 138)(114 139)(115 140)(116 141)(117 142)(118 143)(119 144)(120 133)(157 191)(158 192)(159 181)(160 182)(161 183)(162 184)(163 185)(164 186)(165 187)(166 188)(167 189)(168 190)
(1 17 172 135)(2 136 173 18)(3 19 174 137)(4 138 175 20)(5 21 176 139)(6 140 177 22)(7 23 178 141)(8 142 179 24)(9 13 180 143)(10 144 169 14)(11 15 170 133)(12 134 171 16)(25 158 97 48)(26 37 98 159)(27 160 99 38)(28 39 100 161)(29 162 101 40)(30 41 102 163)(31 164 103 42)(32 43 104 165)(33 166 105 44)(34 45 106 167)(35 168 107 46)(36 47 108 157)(49 111 77 148)(50 149 78 112)(51 113 79 150)(52 151 80 114)(53 115 81 152)(54 153 82 116)(55 117 83 154)(56 155 84 118)(57 119 73 156)(58 145 74 120)(59 109 75 146)(60 147 76 110)(61 95 129 192)(62 181 130 96)(63 85 131 182)(64 183 132 86)(65 87 121 184)(66 185 122 88)(67 89 123 186)(68 187 124 90)(69 91 125 188)(70 189 126 92)(71 93 127 190)(72 191 128 94)
(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)(145 146 147 148 149 150 151 152 153 154 155 156)(157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192)
(1 95 23 123 172 192 141 67)(2 128 142 88 173 72 24 185)(3 93 13 121 174 190 143 65)(4 126 144 86 175 70 14 183)(5 91 15 131 176 188 133 63)(6 124 134 96 177 68 16 181)(7 89 17 129 178 186 135 61)(8 122 136 94 179 66 18 191)(9 87 19 127 180 184 137 71)(10 132 138 92 169 64 20 189)(11 85 21 125 170 182 139 69)(12 130 140 90 171 62 22 187)(25 82 164 110 97 54 42 147)(26 115 43 75 98 152 165 59)(27 80 166 120 99 52 44 145)(28 113 45 73 100 150 167 57)(29 78 168 118 101 50 46 155)(30 111 47 83 102 148 157 55)(31 76 158 116 103 60 48 153)(32 109 37 81 104 146 159 53)(33 74 160 114 105 58 38 151)(34 119 39 79 106 156 161 51)(35 84 162 112 107 56 40 149)(36 117 41 77 108 154 163 49)
G:=sub<Sym(192)| (1,60)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,155)(14,156)(15,145)(16,146)(17,147)(18,148)(19,149)(20,150)(21,151)(22,152)(23,153)(24,154)(25,129)(26,130)(27,131)(28,132)(29,121)(30,122)(31,123)(32,124)(33,125)(34,126)(35,127)(36,128)(37,96)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108)(73,169)(74,170)(75,171)(76,172)(77,173)(78,174)(79,175)(80,176)(81,177)(82,178)(83,179)(84,180)(109,134)(110,135)(111,136)(112,137)(113,138)(114,139)(115,140)(116,141)(117,142)(118,143)(119,144)(120,133)(157,191)(158,192)(159,181)(160,182)(161,183)(162,184)(163,185)(164,186)(165,187)(166,188)(167,189)(168,190), (1,17,172,135)(2,136,173,18)(3,19,174,137)(4,138,175,20)(5,21,176,139)(6,140,177,22)(7,23,178,141)(8,142,179,24)(9,13,180,143)(10,144,169,14)(11,15,170,133)(12,134,171,16)(25,158,97,48)(26,37,98,159)(27,160,99,38)(28,39,100,161)(29,162,101,40)(30,41,102,163)(31,164,103,42)(32,43,104,165)(33,166,105,44)(34,45,106,167)(35,168,107,46)(36,47,108,157)(49,111,77,148)(50,149,78,112)(51,113,79,150)(52,151,80,114)(53,115,81,152)(54,153,82,116)(55,117,83,154)(56,155,84,118)(57,119,73,156)(58,145,74,120)(59,109,75,146)(60,147,76,110)(61,95,129,192)(62,181,130,96)(63,85,131,182)(64,183,132,86)(65,87,121,184)(66,185,122,88)(67,89,123,186)(68,187,124,90)(69,91,125,188)(70,189,126,92)(71,93,127,190)(72,191,128,94), (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)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,95,23,123,172,192,141,67)(2,128,142,88,173,72,24,185)(3,93,13,121,174,190,143,65)(4,126,144,86,175,70,14,183)(5,91,15,131,176,188,133,63)(6,124,134,96,177,68,16,181)(7,89,17,129,178,186,135,61)(8,122,136,94,179,66,18,191)(9,87,19,127,180,184,137,71)(10,132,138,92,169,64,20,189)(11,85,21,125,170,182,139,69)(12,130,140,90,171,62,22,187)(25,82,164,110,97,54,42,147)(26,115,43,75,98,152,165,59)(27,80,166,120,99,52,44,145)(28,113,45,73,100,150,167,57)(29,78,168,118,101,50,46,155)(30,111,47,83,102,148,157,55)(31,76,158,116,103,60,48,153)(32,109,37,81,104,146,159,53)(33,74,160,114,105,58,38,151)(34,119,39,79,106,156,161,51)(35,84,162,112,107,56,40,149)(36,117,41,77,108,154,163,49)>;
G:=Group( (1,60)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,155)(14,156)(15,145)(16,146)(17,147)(18,148)(19,149)(20,150)(21,151)(22,152)(23,153)(24,154)(25,129)(26,130)(27,131)(28,132)(29,121)(30,122)(31,123)(32,124)(33,125)(34,126)(35,127)(36,128)(37,96)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108)(73,169)(74,170)(75,171)(76,172)(77,173)(78,174)(79,175)(80,176)(81,177)(82,178)(83,179)(84,180)(109,134)(110,135)(111,136)(112,137)(113,138)(114,139)(115,140)(116,141)(117,142)(118,143)(119,144)(120,133)(157,191)(158,192)(159,181)(160,182)(161,183)(162,184)(163,185)(164,186)(165,187)(166,188)(167,189)(168,190), (1,17,172,135)(2,136,173,18)(3,19,174,137)(4,138,175,20)(5,21,176,139)(6,140,177,22)(7,23,178,141)(8,142,179,24)(9,13,180,143)(10,144,169,14)(11,15,170,133)(12,134,171,16)(25,158,97,48)(26,37,98,159)(27,160,99,38)(28,39,100,161)(29,162,101,40)(30,41,102,163)(31,164,103,42)(32,43,104,165)(33,166,105,44)(34,45,106,167)(35,168,107,46)(36,47,108,157)(49,111,77,148)(50,149,78,112)(51,113,79,150)(52,151,80,114)(53,115,81,152)(54,153,82,116)(55,117,83,154)(56,155,84,118)(57,119,73,156)(58,145,74,120)(59,109,75,146)(60,147,76,110)(61,95,129,192)(62,181,130,96)(63,85,131,182)(64,183,132,86)(65,87,121,184)(66,185,122,88)(67,89,123,186)(68,187,124,90)(69,91,125,188)(70,189,126,92)(71,93,127,190)(72,191,128,94), (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)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,95,23,123,172,192,141,67)(2,128,142,88,173,72,24,185)(3,93,13,121,174,190,143,65)(4,126,144,86,175,70,14,183)(5,91,15,131,176,188,133,63)(6,124,134,96,177,68,16,181)(7,89,17,129,178,186,135,61)(8,122,136,94,179,66,18,191)(9,87,19,127,180,184,137,71)(10,132,138,92,169,64,20,189)(11,85,21,125,170,182,139,69)(12,130,140,90,171,62,22,187)(25,82,164,110,97,54,42,147)(26,115,43,75,98,152,165,59)(27,80,166,120,99,52,44,145)(28,113,45,73,100,150,167,57)(29,78,168,118,101,50,46,155)(30,111,47,83,102,148,157,55)(31,76,158,116,103,60,48,153)(32,109,37,81,104,146,159,53)(33,74,160,114,105,58,38,151)(34,119,39,79,106,156,161,51)(35,84,162,112,107,56,40,149)(36,117,41,77,108,154,163,49) );
G=PermutationGroup([(1,60),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,55),(9,56),(10,57),(11,58),(12,59),(13,155),(14,156),(15,145),(16,146),(17,147),(18,148),(19,149),(20,150),(21,151),(22,152),(23,153),(24,154),(25,129),(26,130),(27,131),(28,132),(29,121),(30,122),(31,123),(32,124),(33,125),(34,126),(35,127),(36,128),(37,96),(38,85),(39,86),(40,87),(41,88),(42,89),(43,90),(44,91),(45,92),(46,93),(47,94),(48,95),(61,97),(62,98),(63,99),(64,100),(65,101),(66,102),(67,103),(68,104),(69,105),(70,106),(71,107),(72,108),(73,169),(74,170),(75,171),(76,172),(77,173),(78,174),(79,175),(80,176),(81,177),(82,178),(83,179),(84,180),(109,134),(110,135),(111,136),(112,137),(113,138),(114,139),(115,140),(116,141),(117,142),(118,143),(119,144),(120,133),(157,191),(158,192),(159,181),(160,182),(161,183),(162,184),(163,185),(164,186),(165,187),(166,188),(167,189),(168,190)], [(1,17,172,135),(2,136,173,18),(3,19,174,137),(4,138,175,20),(5,21,176,139),(6,140,177,22),(7,23,178,141),(8,142,179,24),(9,13,180,143),(10,144,169,14),(11,15,170,133),(12,134,171,16),(25,158,97,48),(26,37,98,159),(27,160,99,38),(28,39,100,161),(29,162,101,40),(30,41,102,163),(31,164,103,42),(32,43,104,165),(33,166,105,44),(34,45,106,167),(35,168,107,46),(36,47,108,157),(49,111,77,148),(50,149,78,112),(51,113,79,150),(52,151,80,114),(53,115,81,152),(54,153,82,116),(55,117,83,154),(56,155,84,118),(57,119,73,156),(58,145,74,120),(59,109,75,146),(60,147,76,110),(61,95,129,192),(62,181,130,96),(63,85,131,182),(64,183,132,86),(65,87,121,184),(66,185,122,88),(67,89,123,186),(68,187,124,90),(69,91,125,188),(70,189,126,92),(71,93,127,190),(72,191,128,94)], [(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),(145,146,147,148,149,150,151,152,153,154,155,156),(157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192)], [(1,95,23,123,172,192,141,67),(2,128,142,88,173,72,24,185),(3,93,13,121,174,190,143,65),(4,126,144,86,175,70,14,183),(5,91,15,131,176,188,133,63),(6,124,134,96,177,68,16,181),(7,89,17,129,178,186,135,61),(8,122,136,94,179,66,18,191),(9,87,19,127,180,184,137,71),(10,132,138,92,169,64,20,189),(11,85,21,125,170,182,139,69),(12,130,140,90,171,62,22,187),(25,82,164,110,97,54,42,147),(26,115,43,75,98,152,165,59),(27,80,166,120,99,52,44,145),(28,113,45,73,100,150,167,57),(29,78,168,118,101,50,46,155),(30,111,47,83,102,148,157,55),(31,76,158,116,103,60,48,153),(32,109,37,81,104,146,159,53),(33,74,160,114,105,58,38,151),(34,119,39,79,106,156,161,51),(35,84,162,112,107,56,40,149),(36,117,41,77,108,154,163,49)])
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6G | 8A | ··· | 8H | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | D4 | D6 | D6 | SD16 | Dic6 | C4×S3 | C3⋊D4 | C3⋊D4 | D4.S3 | Q8⋊2S3 |
| kernel | C2×C4.Dic6 | C4.Dic6 | C22×C3⋊C8 | C2×C4⋊Dic3 | C6×C4⋊C4 | C2×C3⋊C8 | C2×C4⋊C4 | C2×C12 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 4 | 4 | 2 | 2 | 2 | 2 |
Matrix representation of C2×C4.Dic6 ►in GL5(𝔽73)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 | 2 |
| 0 | 0 | 0 | 72 | 1 |
| 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 61 |
| 0 | 0 | 0 | 6 | 61 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 31 | 70 | 0 | 0 |
| 0 | 28 | 42 | 0 | 0 |
| 0 | 0 | 0 | 0 | 61 |
| 0 | 0 | 0 | 6 | 61 |
G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,2,1],[27,0,0,0,0,0,0,72,0,0,0,1,1,0,0,0,0,0,12,6,0,0,0,61,61],[72,0,0,0,0,0,31,28,0,0,0,70,42,0,0,0,0,0,0,6,0,0,0,61,61] >;
C2×C4.Dic6 in GAP, Magma, Sage, TeX
C_2\times C_4.{\rm Dic}_6 % in TeX
G:=Group("C2xC4.Dic6"); // GroupNames label
G:=SmallGroup(192,522);
// by ID
G=gap.SmallGroup(192,522);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,1094,58,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^12=1,d^2=b*c^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations