metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.28D6, C24.2D10, Dic12⋊2D5, D10.15D12, C120.29C22, C60.122C23, Dic5.17D12, Dic6.20D10, D60.32C22, Dic30.34C22, C6.7(D4×D5), C8.2(S3×D5), C8⋊D5⋊4S3, (C6×D5).4D4, (C4×D5).4D6, C5⋊2C8.2D6, C30.15(C2×D4), C2.12(D5×D12), C10.7(C2×D12), C8⋊D15⋊10C2, C5⋊2(C8.D6), (C5×Dic12)⋊4C2, (D5×Dic6)⋊10C2, C5⋊Dic12⋊11C2, C3⋊1(Q16⋊D5), C15⋊2(C8.C22), (C3×Dic5).4D4, C12.28D10.2C2, Dic6⋊D5⋊11C2, C20.74(C22×S3), (D5×C12).27C22, C12.145(C22×D5), (C5×Dic6).24C22, C4.70(C2×S3×D5), (C3×C8⋊D5)⋊4C2, (C3×C5⋊2C8).20C22, SmallGroup(480,337)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.28D6
G = < a,b,c | a40=b6=1, c2=a20, bab-1=a29, cac-1=a19, cbc-1=b-1 >
Subgroups: 732 in 120 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×4], C22 [×2], C5, S3, C6, C6, C8, C8, C2×C4 [×3], D4 [×2], Q8 [×4], D5 [×2], C10, Dic3 [×3], C12, C12, D6, C2×C6, C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic5, Dic5, C20, C20 [×2], D10, D10, C24, C24, Dic6 [×2], Dic6 [×2], C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D5, D15, C30, C8.C22, C5⋊2C8, C40, Dic10 [×2], C4×D5, C4×D5 [×2], D20 [×2], C5×Q8 [×2], C24⋊C2 [×2], Dic12, Dic12, C3×M4(2), C2×Dic6, C4○D12, C5×Dic3 [×2], C3×Dic5, Dic15, C60, C6×D5, D30, C8⋊D5, C40⋊C2, Q8⋊D5, C5⋊Q16, C5×Q16, Q8×D5, Q8⋊2D5, C8.D6, C3×C5⋊2C8, C120, D5×Dic3, D30.C2, C3⋊D20, C15⋊Q8, D5×C12, C5×Dic6 [×2], Dic30, D60, Q16⋊D5, Dic6⋊D5, C5⋊Dic12, C3×C8⋊D5, C5×Dic12, C8⋊D15, D5×Dic6, C12.28D10, C40.28D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], D12 [×2], C22×S3, C8.C22, C22×D5, C2×D12, S3×D5, D4×D5, C8.D6, C2×S3×D5, Q16⋊D5, D5×D12, C40.28D6
►On 240 points
Generators in S240
(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)(201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)
(1 70 149)(2 59 150 30 71 138)(3 48 151 19 72 127)(4 77 152 8 73 156)(5 66 153 37 74 145)(6 55 154 26 75 134)(7 44 155 15 76 123)(9 62 157 33 78 141)(10 51 158 22 79 130)(11 80 159)(12 69 160 40 41 148)(13 58 121 29 42 137)(14 47 122 18 43 126)(16 65 124 36 45 144)(17 54 125 25 46 133)(20 61 128 32 49 140)(21 50 129)(23 68 131 39 52 147)(24 57 132 28 53 136)(27 64 135 35 56 143)(31 60 139)(34 67 142 38 63 146)(81 229 183 113 237 175)(82 218 184 102 238 164)(83 207 185 91 239 193)(84 236 186 120 240 182)(85 225 187 109 201 171)(86 214 188 98 202 200)(87 203 189)(88 232 190 116 204 178)(89 221 191 105 205 167)(90 210 192 94 206 196)(92 228 194 112 208 174)(93 217 195 101 209 163)(95 235 197 119 211 181)(96 224 198 108 212 170)(97 213 199)(99 231 161 115 215 177)(100 220 162 104 216 166)(103 227 165 111 219 173)(106 234 168 118 222 180)(107 223 169)(110 230 172 114 226 176)(117 233 179)
(1 199 21 179)(2 178 22 198)(3 197 23 177)(4 176 24 196)(5 195 25 175)(6 174 26 194)(7 193 27 173)(8 172 28 192)(9 191 29 171)(10 170 30 190)(11 189 31 169)(12 168 32 188)(13 187 33 167)(14 166 34 186)(15 185 35 165)(16 164 36 184)(17 183 37 163)(18 162 38 182)(19 181 39 161)(20 200 40 180)(41 222 61 202)(42 201 62 221)(43 220 63 240)(44 239 64 219)(45 218 65 238)(46 237 66 217)(47 216 67 236)(48 235 68 215)(49 214 69 234)(50 233 70 213)(51 212 71 232)(52 231 72 211)(53 210 73 230)(54 229 74 209)(55 208 75 228)(56 227 76 207)(57 206 77 226)(58 225 78 205)(59 204 79 224)(60 223 80 203)(81 145 101 125)(82 124 102 144)(83 143 103 123)(84 122 104 142)(85 141 105 121)(86 160 106 140)(87 139 107 159)(88 158 108 138)(89 137 109 157)(90 156 110 136)(91 135 111 155)(92 154 112 134)(93 133 113 153)(94 152 114 132)(95 131 115 151)(96 150 116 130)(97 129 117 149)(98 148 118 128)(99 127 119 147)(100 146 120 126)
G:=sub<Sym(240)| (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)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,70,149)(2,59,150,30,71,138)(3,48,151,19,72,127)(4,77,152,8,73,156)(5,66,153,37,74,145)(6,55,154,26,75,134)(7,44,155,15,76,123)(9,62,157,33,78,141)(10,51,158,22,79,130)(11,80,159)(12,69,160,40,41,148)(13,58,121,29,42,137)(14,47,122,18,43,126)(16,65,124,36,45,144)(17,54,125,25,46,133)(20,61,128,32,49,140)(21,50,129)(23,68,131,39,52,147)(24,57,132,28,53,136)(27,64,135,35,56,143)(31,60,139)(34,67,142,38,63,146)(81,229,183,113,237,175)(82,218,184,102,238,164)(83,207,185,91,239,193)(84,236,186,120,240,182)(85,225,187,109,201,171)(86,214,188,98,202,200)(87,203,189)(88,232,190,116,204,178)(89,221,191,105,205,167)(90,210,192,94,206,196)(92,228,194,112,208,174)(93,217,195,101,209,163)(95,235,197,119,211,181)(96,224,198,108,212,170)(97,213,199)(99,231,161,115,215,177)(100,220,162,104,216,166)(103,227,165,111,219,173)(106,234,168,118,222,180)(107,223,169)(110,230,172,114,226,176)(117,233,179), (1,199,21,179)(2,178,22,198)(3,197,23,177)(4,176,24,196)(5,195,25,175)(6,174,26,194)(7,193,27,173)(8,172,28,192)(9,191,29,171)(10,170,30,190)(11,189,31,169)(12,168,32,188)(13,187,33,167)(14,166,34,186)(15,185,35,165)(16,164,36,184)(17,183,37,163)(18,162,38,182)(19,181,39,161)(20,200,40,180)(41,222,61,202)(42,201,62,221)(43,220,63,240)(44,239,64,219)(45,218,65,238)(46,237,66,217)(47,216,67,236)(48,235,68,215)(49,214,69,234)(50,233,70,213)(51,212,71,232)(52,231,72,211)(53,210,73,230)(54,229,74,209)(55,208,75,228)(56,227,76,207)(57,206,77,226)(58,225,78,205)(59,204,79,224)(60,223,80,203)(81,145,101,125)(82,124,102,144)(83,143,103,123)(84,122,104,142)(85,141,105,121)(86,160,106,140)(87,139,107,159)(88,158,108,138)(89,137,109,157)(90,156,110,136)(91,135,111,155)(92,154,112,134)(93,133,113,153)(94,152,114,132)(95,131,115,151)(96,150,116,130)(97,129,117,149)(98,148,118,128)(99,127,119,147)(100,146,120,126)>;
G:=Group( (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)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,70,149)(2,59,150,30,71,138)(3,48,151,19,72,127)(4,77,152,8,73,156)(5,66,153,37,74,145)(6,55,154,26,75,134)(7,44,155,15,76,123)(9,62,157,33,78,141)(10,51,158,22,79,130)(11,80,159)(12,69,160,40,41,148)(13,58,121,29,42,137)(14,47,122,18,43,126)(16,65,124,36,45,144)(17,54,125,25,46,133)(20,61,128,32,49,140)(21,50,129)(23,68,131,39,52,147)(24,57,132,28,53,136)(27,64,135,35,56,143)(31,60,139)(34,67,142,38,63,146)(81,229,183,113,237,175)(82,218,184,102,238,164)(83,207,185,91,239,193)(84,236,186,120,240,182)(85,225,187,109,201,171)(86,214,188,98,202,200)(87,203,189)(88,232,190,116,204,178)(89,221,191,105,205,167)(90,210,192,94,206,196)(92,228,194,112,208,174)(93,217,195,101,209,163)(95,235,197,119,211,181)(96,224,198,108,212,170)(97,213,199)(99,231,161,115,215,177)(100,220,162,104,216,166)(103,227,165,111,219,173)(106,234,168,118,222,180)(107,223,169)(110,230,172,114,226,176)(117,233,179), (1,199,21,179)(2,178,22,198)(3,197,23,177)(4,176,24,196)(5,195,25,175)(6,174,26,194)(7,193,27,173)(8,172,28,192)(9,191,29,171)(10,170,30,190)(11,189,31,169)(12,168,32,188)(13,187,33,167)(14,166,34,186)(15,185,35,165)(16,164,36,184)(17,183,37,163)(18,162,38,182)(19,181,39,161)(20,200,40,180)(41,222,61,202)(42,201,62,221)(43,220,63,240)(44,239,64,219)(45,218,65,238)(46,237,66,217)(47,216,67,236)(48,235,68,215)(49,214,69,234)(50,233,70,213)(51,212,71,232)(52,231,72,211)(53,210,73,230)(54,229,74,209)(55,208,75,228)(56,227,76,207)(57,206,77,226)(58,225,78,205)(59,204,79,224)(60,223,80,203)(81,145,101,125)(82,124,102,144)(83,143,103,123)(84,122,104,142)(85,141,105,121)(86,160,106,140)(87,139,107,159)(88,158,108,138)(89,137,109,157)(90,156,110,136)(91,135,111,155)(92,154,112,134)(93,133,113,153)(94,152,114,132)(95,131,115,151)(96,150,116,130)(97,129,117,149)(98,148,118,128)(99,127,119,147)(100,146,120,126) );
G=PermutationGroup([(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),(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)], [(1,70,149),(2,59,150,30,71,138),(3,48,151,19,72,127),(4,77,152,8,73,156),(5,66,153,37,74,145),(6,55,154,26,75,134),(7,44,155,15,76,123),(9,62,157,33,78,141),(10,51,158,22,79,130),(11,80,159),(12,69,160,40,41,148),(13,58,121,29,42,137),(14,47,122,18,43,126),(16,65,124,36,45,144),(17,54,125,25,46,133),(20,61,128,32,49,140),(21,50,129),(23,68,131,39,52,147),(24,57,132,28,53,136),(27,64,135,35,56,143),(31,60,139),(34,67,142,38,63,146),(81,229,183,113,237,175),(82,218,184,102,238,164),(83,207,185,91,239,193),(84,236,186,120,240,182),(85,225,187,109,201,171),(86,214,188,98,202,200),(87,203,189),(88,232,190,116,204,178),(89,221,191,105,205,167),(90,210,192,94,206,196),(92,228,194,112,208,174),(93,217,195,101,209,163),(95,235,197,119,211,181),(96,224,198,108,212,170),(97,213,199),(99,231,161,115,215,177),(100,220,162,104,216,166),(103,227,165,111,219,173),(106,234,168,118,222,180),(107,223,169),(110,230,172,114,226,176),(117,233,179)], [(1,199,21,179),(2,178,22,198),(3,197,23,177),(4,176,24,196),(5,195,25,175),(6,174,26,194),(7,193,27,173),(8,172,28,192),(9,191,29,171),(10,170,30,190),(11,189,31,169),(12,168,32,188),(13,187,33,167),(14,166,34,186),(15,185,35,165),(16,164,36,184),(17,183,37,163),(18,162,38,182),(19,181,39,161),(20,200,40,180),(41,222,61,202),(42,201,62,221),(43,220,63,240),(44,239,64,219),(45,218,65,238),(46,237,66,217),(47,216,67,236),(48,235,68,215),(49,214,69,234),(50,233,70,213),(51,212,71,232),(52,231,72,211),(53,210,73,230),(54,229,74,209),(55,208,75,228),(56,227,76,207),(57,206,77,226),(58,225,78,205),(59,204,79,224),(60,223,80,203),(81,145,101,125),(82,124,102,144),(83,143,103,123),(84,122,104,142),(85,141,105,121),(86,160,106,140),(87,139,107,159),(88,158,108,138),(89,137,109,157),(90,156,110,136),(91,135,111,155),(92,154,112,134),(93,133,113,153),(94,152,114,132),(95,131,115,151),(96,150,116,130),(97,129,117,149),(98,148,118,128),(99,127,119,147),(100,146,120,126)])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 8A | 8B | 10A | 10B | 12A | 12B | 12C | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 24C | 24D | 30A | 30B | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 10 | 60 | 2 | 2 | 10 | 12 | 12 | 60 | 2 | 2 | 2 | 20 | 4 | 20 | 2 | 2 | 2 | 2 | 20 | 4 | 4 | 4 | 4 | 24 | 24 | 24 | 24 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D12 | D12 | C8.C22 | S3×D5 | D4×D5 | C8.D6 | C2×S3×D5 | Q16⋊D5 | D5×D12 | C40.28D6 |
| kernel | C40.28D6 | Dic6⋊D5 | C5⋊Dic12 | C3×C8⋊D5 | C5×Dic12 | C8⋊D15 | D5×Dic6 | C12.28D10 | C8⋊D5 | C3×Dic5 | C6×D5 | Dic12 | C5⋊2C8 | C40 | C4×D5 | C24 | Dic6 | Dic5 | D10 | C15 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C40.28D6 ►in GL8(𝔽241)
| 0 | 12 | 0 | 24 | 0 | 0 | 0 | 0 |
| 87 | 87 | 174 | 174 | 0 | 0 | 0 | 0 |
| 0 | 217 | 0 | 229 | 0 | 0 | 0 | 0 |
| 67 | 67 | 154 | 154 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 142 | 98 | 5 | 0 |
| 0 | 0 | 0 | 0 | 143 | 44 | 0 | 5 |
| 0 | 0 | 0 | 0 | 125 | 9 | 99 | 143 |
| 0 | 0 | 0 | 0 | 232 | 116 | 98 | 197 |
| 52 | 51 | 52 | 51 | 0 | 0 | 0 | 0 |
| 188 | 189 | 188 | 189 | 0 | 0 | 0 | 0 |
| 189 | 190 | 0 | 0 | 0 | 0 | 0 | 0 |
| 53 | 52 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 114 | 105 | 1 | 1 |
| 0 | 0 | 0 | 0 | 136 | 9 | 240 | 0 |
| 77 | 6 | 192 | 40 | 0 | 0 | 0 | 0 |
| 93 | 164 | 138 | 49 | 0 | 0 | 0 | 0 |
| 115 | 34 | 164 | 235 | 0 | 0 | 0 | 0 |
| 45 | 126 | 148 | 77 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 148 | 95 | 0 | 0 |
| 0 | 0 | 0 | 0 | 188 | 93 | 0 | 0 |
| 0 | 0 | 0 | 0 | 130 | 20 | 180 | 137 |
| 0 | 0 | 0 | 0 | 131 | 111 | 198 | 61 |
G:=sub<GL(8,GF(241))| [0,87,0,67,0,0,0,0,12,87,217,67,0,0,0,0,0,174,0,154,0,0,0,0,24,174,229,154,0,0,0,0,0,0,0,0,142,143,125,232,0,0,0,0,98,44,9,116,0,0,0,0,5,0,99,98,0,0,0,0,0,5,143,197],[52,188,189,53,0,0,0,0,51,189,190,52,0,0,0,0,52,188,0,0,0,0,0,0,51,189,0,0,0,0,0,0,0,0,0,0,240,1,114,136,0,0,0,0,240,0,105,9,0,0,0,0,0,0,1,240,0,0,0,0,0,0,1,0],[77,93,115,45,0,0,0,0,6,164,34,126,0,0,0,0,192,138,164,148,0,0,0,0,40,49,235,77,0,0,0,0,0,0,0,0,148,188,130,131,0,0,0,0,95,93,20,111,0,0,0,0,0,0,180,198,0,0,0,0,0,0,137,61] >;
C40.28D6 in GAP, Magma, Sage, TeX
C_{40}._{28}D_6 % in TeX
G:=Group("C40.28D6"); // GroupNames label
G:=SmallGroup(480,337);
// by ID
G=gap.SmallGroup(480,337);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,422,135,142,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^40=b^6=1,c^2=a^20,b*a*b^-1=a^29,c*a*c^-1=a^19,c*b*c^-1=b^-1>;
// generators/relations