metabelian, supersoluble, monomial, A-group
Aliases: C33⋊7C8, C32⋊6(C3⋊C8), C3⋊(C32⋊4C8), C12.6(C3⋊S3), (C3×C12).20S3, C2.(C33⋊5C4), (C32×C6).5C4, (C32×C12).6C2, C6.3(C3⋊Dic3), (C3×C6).11Dic3, C4.2(C33⋊C2), SmallGroup(216,84)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — C33⋊7C8 |
Generators and relations for C33⋊7C8
G = < a,b,c,d | a3=b3=c3=d8=1, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 268 in 112 conjugacy classes, 85 normal (7 characteristic)
C1, C2, C3, C4, C6, C8, C32, C12, C3×C6, C3⋊C8, C33, C3×C12, C32×C6, C32⋊4C8, C32×C12, C33⋊7C8
Quotients: C1, C2, C4, S3, C8, Dic3, C3⋊S3, C3⋊C8, C3⋊Dic3, C33⋊C2, C32⋊4C8, C33⋊5C4, C33⋊7C8
►Regular action on 216 points
Generators in S216
(1 162 68)(2 69 163)(3 164 70)(4 71 165)(5 166 72)(6 65 167)(7 168 66)(8 67 161)(9 107 20)(10 21 108)(11 109 22)(12 23 110)(13 111 24)(14 17 112)(15 105 18)(16 19 106)(25 59 181)(26 182 60)(27 61 183)(28 184 62)(29 63 177)(30 178 64)(31 57 179)(32 180 58)(33 53 127)(34 128 54)(35 55 121)(36 122 56)(37 49 123)(38 124 50)(39 51 125)(40 126 52)(41 133 146)(42 147 134)(43 135 148)(44 149 136)(45 129 150)(46 151 130)(47 131 152)(48 145 132)(73 143 210)(74 211 144)(75 137 212)(76 213 138)(77 139 214)(78 215 140)(79 141 216)(80 209 142)(81 113 155)(82 156 114)(83 115 157)(84 158 116)(85 117 159)(86 160 118)(87 119 153)(88 154 120)(89 172 208)(90 201 173)(91 174 202)(92 203 175)(93 176 204)(94 205 169)(95 170 206)(96 207 171)(97 189 193)(98 194 190)(99 191 195)(100 196 192)(101 185 197)(102 198 186)(103 187 199)(104 200 188)
(1 76 47)(2 48 77)(3 78 41)(4 42 79)(5 80 43)(6 44 73)(7 74 45)(8 46 75)(9 172 116)(10 117 173)(11 174 118)(12 119 175)(13 176 120)(14 113 169)(15 170 114)(16 115 171)(17 155 94)(18 95 156)(19 157 96)(20 89 158)(21 159 90)(22 91 160)(23 153 92)(24 93 154)(25 56 99)(26 100 49)(27 50 101)(28 102 51)(29 52 103)(30 104 53)(31 54 97)(32 98 55)(33 64 188)(34 189 57)(35 58 190)(36 191 59)(37 60 192)(38 185 61)(39 62 186)(40 187 63)(65 149 143)(66 144 150)(67 151 137)(68 138 152)(69 145 139)(70 140 146)(71 147 141)(72 142 148)(81 205 112)(82 105 206)(83 207 106)(84 107 208)(85 201 108)(86 109 202)(87 203 110)(88 111 204)(121 180 194)(122 195 181)(123 182 196)(124 197 183)(125 184 198)(126 199 177)(127 178 200)(128 193 179)(129 168 211)(130 212 161)(131 162 213)(132 214 163)(133 164 215)(134 216 165)(135 166 209)(136 210 167)
(1 54 113)(2 114 55)(3 56 115)(4 116 49)(5 50 117)(6 118 51)(7 52 119)(8 120 53)(9 26 42)(10 43 27)(11 28 44)(12 45 29)(13 30 46)(14 47 31)(15 32 48)(16 41 25)(17 131 57)(18 58 132)(19 133 59)(20 60 134)(21 135 61)(22 62 136)(23 129 63)(24 64 130)(33 161 154)(34 155 162)(35 163 156)(36 157 164)(37 165 158)(38 159 166)(39 167 160)(40 153 168)(65 86 125)(66 126 87)(67 88 127)(68 128 81)(69 82 121)(70 122 83)(71 84 123)(72 124 85)(73 174 102)(74 103 175)(75 176 104)(76 97 169)(77 170 98)(78 99 171)(79 172 100)(80 101 173)(89 192 216)(90 209 185)(91 186 210)(92 211 187)(93 188 212)(94 213 189)(95 190 214)(96 215 191)(105 180 145)(106 146 181)(107 182 147)(108 148 183)(109 184 149)(110 150 177)(111 178 151)(112 152 179)(137 204 200)(138 193 205)(139 206 194)(140 195 207)(141 208 196)(142 197 201)(143 202 198)(144 199 203)
(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)(193 194 195 196 197 198 199 200)(201 202 203 204 205 206 207 208)(209 210 211 212 213 214 215 216)
G:=sub<Sym(216)| (1,162,68)(2,69,163)(3,164,70)(4,71,165)(5,166,72)(6,65,167)(7,168,66)(8,67,161)(9,107,20)(10,21,108)(11,109,22)(12,23,110)(13,111,24)(14,17,112)(15,105,18)(16,19,106)(25,59,181)(26,182,60)(27,61,183)(28,184,62)(29,63,177)(30,178,64)(31,57,179)(32,180,58)(33,53,127)(34,128,54)(35,55,121)(36,122,56)(37,49,123)(38,124,50)(39,51,125)(40,126,52)(41,133,146)(42,147,134)(43,135,148)(44,149,136)(45,129,150)(46,151,130)(47,131,152)(48,145,132)(73,143,210)(74,211,144)(75,137,212)(76,213,138)(77,139,214)(78,215,140)(79,141,216)(80,209,142)(81,113,155)(82,156,114)(83,115,157)(84,158,116)(85,117,159)(86,160,118)(87,119,153)(88,154,120)(89,172,208)(90,201,173)(91,174,202)(92,203,175)(93,176,204)(94,205,169)(95,170,206)(96,207,171)(97,189,193)(98,194,190)(99,191,195)(100,196,192)(101,185,197)(102,198,186)(103,187,199)(104,200,188), (1,76,47)(2,48,77)(3,78,41)(4,42,79)(5,80,43)(6,44,73)(7,74,45)(8,46,75)(9,172,116)(10,117,173)(11,174,118)(12,119,175)(13,176,120)(14,113,169)(15,170,114)(16,115,171)(17,155,94)(18,95,156)(19,157,96)(20,89,158)(21,159,90)(22,91,160)(23,153,92)(24,93,154)(25,56,99)(26,100,49)(27,50,101)(28,102,51)(29,52,103)(30,104,53)(31,54,97)(32,98,55)(33,64,188)(34,189,57)(35,58,190)(36,191,59)(37,60,192)(38,185,61)(39,62,186)(40,187,63)(65,149,143)(66,144,150)(67,151,137)(68,138,152)(69,145,139)(70,140,146)(71,147,141)(72,142,148)(81,205,112)(82,105,206)(83,207,106)(84,107,208)(85,201,108)(86,109,202)(87,203,110)(88,111,204)(121,180,194)(122,195,181)(123,182,196)(124,197,183)(125,184,198)(126,199,177)(127,178,200)(128,193,179)(129,168,211)(130,212,161)(131,162,213)(132,214,163)(133,164,215)(134,216,165)(135,166,209)(136,210,167), (1,54,113)(2,114,55)(3,56,115)(4,116,49)(5,50,117)(6,118,51)(7,52,119)(8,120,53)(9,26,42)(10,43,27)(11,28,44)(12,45,29)(13,30,46)(14,47,31)(15,32,48)(16,41,25)(17,131,57)(18,58,132)(19,133,59)(20,60,134)(21,135,61)(22,62,136)(23,129,63)(24,64,130)(33,161,154)(34,155,162)(35,163,156)(36,157,164)(37,165,158)(38,159,166)(39,167,160)(40,153,168)(65,86,125)(66,126,87)(67,88,127)(68,128,81)(69,82,121)(70,122,83)(71,84,123)(72,124,85)(73,174,102)(74,103,175)(75,176,104)(76,97,169)(77,170,98)(78,99,171)(79,172,100)(80,101,173)(89,192,216)(90,209,185)(91,186,210)(92,211,187)(93,188,212)(94,213,189)(95,190,214)(96,215,191)(105,180,145)(106,146,181)(107,182,147)(108,148,183)(109,184,149)(110,150,177)(111,178,151)(112,152,179)(137,204,200)(138,193,205)(139,206,194)(140,195,207)(141,208,196)(142,197,201)(143,202,198)(144,199,203), (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)(193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208)(209,210,211,212,213,214,215,216)>;
G:=Group( (1,162,68)(2,69,163)(3,164,70)(4,71,165)(5,166,72)(6,65,167)(7,168,66)(8,67,161)(9,107,20)(10,21,108)(11,109,22)(12,23,110)(13,111,24)(14,17,112)(15,105,18)(16,19,106)(25,59,181)(26,182,60)(27,61,183)(28,184,62)(29,63,177)(30,178,64)(31,57,179)(32,180,58)(33,53,127)(34,128,54)(35,55,121)(36,122,56)(37,49,123)(38,124,50)(39,51,125)(40,126,52)(41,133,146)(42,147,134)(43,135,148)(44,149,136)(45,129,150)(46,151,130)(47,131,152)(48,145,132)(73,143,210)(74,211,144)(75,137,212)(76,213,138)(77,139,214)(78,215,140)(79,141,216)(80,209,142)(81,113,155)(82,156,114)(83,115,157)(84,158,116)(85,117,159)(86,160,118)(87,119,153)(88,154,120)(89,172,208)(90,201,173)(91,174,202)(92,203,175)(93,176,204)(94,205,169)(95,170,206)(96,207,171)(97,189,193)(98,194,190)(99,191,195)(100,196,192)(101,185,197)(102,198,186)(103,187,199)(104,200,188), (1,76,47)(2,48,77)(3,78,41)(4,42,79)(5,80,43)(6,44,73)(7,74,45)(8,46,75)(9,172,116)(10,117,173)(11,174,118)(12,119,175)(13,176,120)(14,113,169)(15,170,114)(16,115,171)(17,155,94)(18,95,156)(19,157,96)(20,89,158)(21,159,90)(22,91,160)(23,153,92)(24,93,154)(25,56,99)(26,100,49)(27,50,101)(28,102,51)(29,52,103)(30,104,53)(31,54,97)(32,98,55)(33,64,188)(34,189,57)(35,58,190)(36,191,59)(37,60,192)(38,185,61)(39,62,186)(40,187,63)(65,149,143)(66,144,150)(67,151,137)(68,138,152)(69,145,139)(70,140,146)(71,147,141)(72,142,148)(81,205,112)(82,105,206)(83,207,106)(84,107,208)(85,201,108)(86,109,202)(87,203,110)(88,111,204)(121,180,194)(122,195,181)(123,182,196)(124,197,183)(125,184,198)(126,199,177)(127,178,200)(128,193,179)(129,168,211)(130,212,161)(131,162,213)(132,214,163)(133,164,215)(134,216,165)(135,166,209)(136,210,167), (1,54,113)(2,114,55)(3,56,115)(4,116,49)(5,50,117)(6,118,51)(7,52,119)(8,120,53)(9,26,42)(10,43,27)(11,28,44)(12,45,29)(13,30,46)(14,47,31)(15,32,48)(16,41,25)(17,131,57)(18,58,132)(19,133,59)(20,60,134)(21,135,61)(22,62,136)(23,129,63)(24,64,130)(33,161,154)(34,155,162)(35,163,156)(36,157,164)(37,165,158)(38,159,166)(39,167,160)(40,153,168)(65,86,125)(66,126,87)(67,88,127)(68,128,81)(69,82,121)(70,122,83)(71,84,123)(72,124,85)(73,174,102)(74,103,175)(75,176,104)(76,97,169)(77,170,98)(78,99,171)(79,172,100)(80,101,173)(89,192,216)(90,209,185)(91,186,210)(92,211,187)(93,188,212)(94,213,189)(95,190,214)(96,215,191)(105,180,145)(106,146,181)(107,182,147)(108,148,183)(109,184,149)(110,150,177)(111,178,151)(112,152,179)(137,204,200)(138,193,205)(139,206,194)(140,195,207)(141,208,196)(142,197,201)(143,202,198)(144,199,203), (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)(193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208)(209,210,211,212,213,214,215,216) );
G=PermutationGroup([[(1,162,68),(2,69,163),(3,164,70),(4,71,165),(5,166,72),(6,65,167),(7,168,66),(8,67,161),(9,107,20),(10,21,108),(11,109,22),(12,23,110),(13,111,24),(14,17,112),(15,105,18),(16,19,106),(25,59,181),(26,182,60),(27,61,183),(28,184,62),(29,63,177),(30,178,64),(31,57,179),(32,180,58),(33,53,127),(34,128,54),(35,55,121),(36,122,56),(37,49,123),(38,124,50),(39,51,125),(40,126,52),(41,133,146),(42,147,134),(43,135,148),(44,149,136),(45,129,150),(46,151,130),(47,131,152),(48,145,132),(73,143,210),(74,211,144),(75,137,212),(76,213,138),(77,139,214),(78,215,140),(79,141,216),(80,209,142),(81,113,155),(82,156,114),(83,115,157),(84,158,116),(85,117,159),(86,160,118),(87,119,153),(88,154,120),(89,172,208),(90,201,173),(91,174,202),(92,203,175),(93,176,204),(94,205,169),(95,170,206),(96,207,171),(97,189,193),(98,194,190),(99,191,195),(100,196,192),(101,185,197),(102,198,186),(103,187,199),(104,200,188)], [(1,76,47),(2,48,77),(3,78,41),(4,42,79),(5,80,43),(6,44,73),(7,74,45),(8,46,75),(9,172,116),(10,117,173),(11,174,118),(12,119,175),(13,176,120),(14,113,169),(15,170,114),(16,115,171),(17,155,94),(18,95,156),(19,157,96),(20,89,158),(21,159,90),(22,91,160),(23,153,92),(24,93,154),(25,56,99),(26,100,49),(27,50,101),(28,102,51),(29,52,103),(30,104,53),(31,54,97),(32,98,55),(33,64,188),(34,189,57),(35,58,190),(36,191,59),(37,60,192),(38,185,61),(39,62,186),(40,187,63),(65,149,143),(66,144,150),(67,151,137),(68,138,152),(69,145,139),(70,140,146),(71,147,141),(72,142,148),(81,205,112),(82,105,206),(83,207,106),(84,107,208),(85,201,108),(86,109,202),(87,203,110),(88,111,204),(121,180,194),(122,195,181),(123,182,196),(124,197,183),(125,184,198),(126,199,177),(127,178,200),(128,193,179),(129,168,211),(130,212,161),(131,162,213),(132,214,163),(133,164,215),(134,216,165),(135,166,209),(136,210,167)], [(1,54,113),(2,114,55),(3,56,115),(4,116,49),(5,50,117),(6,118,51),(7,52,119),(8,120,53),(9,26,42),(10,43,27),(11,28,44),(12,45,29),(13,30,46),(14,47,31),(15,32,48),(16,41,25),(17,131,57),(18,58,132),(19,133,59),(20,60,134),(21,135,61),(22,62,136),(23,129,63),(24,64,130),(33,161,154),(34,155,162),(35,163,156),(36,157,164),(37,165,158),(38,159,166),(39,167,160),(40,153,168),(65,86,125),(66,126,87),(67,88,127),(68,128,81),(69,82,121),(70,122,83),(71,84,123),(72,124,85),(73,174,102),(74,103,175),(75,176,104),(76,97,169),(77,170,98),(78,99,171),(79,172,100),(80,101,173),(89,192,216),(90,209,185),(91,186,210),(92,211,187),(93,188,212),(94,213,189),(95,190,214),(96,215,191),(105,180,145),(106,146,181),(107,182,147),(108,148,183),(109,184,149),(110,150,177),(111,178,151),(112,152,179),(137,204,200),(138,193,205),(139,206,194),(140,195,207),(141,208,196),(142,197,201),(143,202,198),(144,199,203)], [(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),(193,194,195,196,197,198,199,200),(201,202,203,204,205,206,207,208),(209,210,211,212,213,214,215,216)]])
C33⋊7C8 is a maximal subgroup of
S3×C32⋊4C8 C3⋊S3×C3⋊C8 C33⋊7M4(2) C33⋊8M4(2) C33⋊6D8 C33⋊12SD16 C33⋊13SD16 C33⋊6Q16 C8×C33⋊C2 C33⋊15M4(2) C33⋊18M4(2) C33⋊15D8 C33⋊24SD16 C33⋊27SD16 C33⋊15Q16
C33⋊7C8 is a maximal quotient of
C33⋊7C16
60 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3M | 4A | 4B | 6A | ··· | 6M | 8A | 8B | 8C | 8D | 12A | ··· | 12Z |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 2 | ··· | 2 | 27 | 27 | 27 | 27 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | - | |||
| image | C1 | C2 | C4 | C8 | S3 | Dic3 | C3⋊C8 |
| kernel | C33⋊7C8 | C32×C12 | C32×C6 | C33 | C3×C12 | C3×C6 | C32 |
| # reps | 1 | 1 | 2 | 4 | 13 | 13 | 26 |
Matrix representation of C33⋊7C8 ►in GL6(𝔽73)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 59 | 54 | 0 | 0 | 0 | 0 |
| 68 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 59 | 0 | 0 |
| 0 | 0 | 54 | 68 | 0 | 0 |
| 0 | 0 | 0 | 0 | 70 | 70 |
| 0 | 0 | 0 | 0 | 67 | 3 |
G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[59,68,0,0,0,0,54,14,0,0,0,0,0,0,5,54,0,0,0,0,59,68,0,0,0,0,0,0,70,67,0,0,0,0,70,3] >;
C33⋊7C8 in GAP, Magma, Sage, TeX
C_3^3\rtimes_7C_8
% in TeX
G:=Group("C3^3:7C8"); // GroupNames label
G:=SmallGroup(216,84);
// by ID
G=gap.SmallGroup(216,84);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,12,31,387,1444,5189]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations