metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C7⋊C8⋊26D4, C7⋊3(C8⋊9D4), D14⋊C8⋊21C2, C22⋊C8⋊15D7, C56⋊C4⋊14C2, C4.199(D4×D7), C14.29(C4×D4), Dic7⋊C8⋊21C2, D14⋊C4.10C4, (C2×C8).164D14, C28.358(C2×D4), (C2×C14)⋊2M4(2), C23.24(C4×D7), C23.D7.7C4, C14.25(C8○D4), C22⋊1(C8⋊D7), Dic7⋊C4.10C4, C14.6(C2×M4(2)), C28.300(C4○D4), (C2×C28).825C23, (C2×C56).174C22, C2.11(D28.C4), (C22×C4).305D14, C4.126(D4⋊2D7), C2.13(Dic7⋊4D4), (C22×C28).339C22, (C4×Dic7).183C22, (C22×C7⋊C8)⋊17C2, (C2×C4).64(C4×D7), (C2×C7⋊D4).7C4, (C2×C8⋊D7)⋊14C2, (C7×C22⋊C8)⋊19C2, C2.10(C2×C8⋊D7), (C4×C7⋊D4).14C2, C22.107(C2×C4×D7), (C2×C28).155(C2×C4), (C2×C7⋊C8).302C22, (C2×C4×D7).180C22, (C22×C14).43(C2×C4), (C2×C14).80(C22×C4), (C2×Dic7).18(C2×C4), (C22×D7).14(C2×C4), (C2×C4).767(C22×D7), SmallGroup(448,264)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7⋊C8⋊26D4
G = < a,b,c,d | a7=b8=c4=d2=1, bab-1=cac-1=a-1, ad=da, cbc-1=b5, bd=db, dcd=c-1 >
Subgroups: 508 in 124 conjugacy classes, 53 normal (47 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C8⋊C4, C22⋊C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), C7⋊C8, C7⋊C8, C56, C4×D7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C8⋊9D4, C8⋊D7, C2×C7⋊C8, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, D14⋊C4, C23.D7, C2×C56, C2×C4×D7, C2×C7⋊D4, C22×C28, Dic7⋊C8, C56⋊C4, D14⋊C8, C7×C22⋊C8, C2×C8⋊D7, C22×C7⋊C8, C4×C7⋊D4, C7⋊C8⋊26D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, M4(2), C22×C4, C2×D4, C4○D4, D14, C4×D4, C2×M4(2), C8○D4, C4×D7, C22×D7, C8⋊9D4, C8⋊D7, C2×C4×D7, D4×D7, D4⋊2D7, Dic7⋊4D4, C2×C8⋊D7, D28.C4, C7⋊C8⋊26D4
►On 224 points
Generators in S224
(1 219 187 122 119 182 214)(2 215 183 120 123 188 220)(3 221 189 124 113 184 216)(4 209 177 114 125 190 222)(5 223 191 126 115 178 210)(6 211 179 116 127 192 224)(7 217 185 128 117 180 212)(8 213 181 118 121 186 218)(9 154 20 170 144 133 195)(10 196 134 137 171 21 155)(11 156 22 172 138 135 197)(12 198 136 139 173 23 157)(13 158 24 174 140 129 199)(14 200 130 141 175 17 159)(15 160 18 176 142 131 193)(16 194 132 143 169 19 153)(25 36 107 76 48 147 81)(26 82 148 41 77 108 37)(27 38 109 78 42 149 83)(28 84 150 43 79 110 39)(29 40 111 80 44 151 85)(30 86 152 45 73 112 33)(31 34 105 74 46 145 87)(32 88 146 47 75 106 35)(49 207 91 66 161 57 104)(50 97 58 162 67 92 208)(51 201 93 68 163 59 98)(52 99 60 164 69 94 202)(53 203 95 70 165 61 100)(54 101 62 166 71 96 204)(55 205 89 72 167 63 102)(56 103 64 168 65 90 206)
(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)
(1 153 47 164)(2 158 48 161)(3 155 41 166)(4 160 42 163)(5 157 43 168)(6 154 44 165)(7 159 45 162)(8 156 46 167)(9 151 70 211)(10 148 71 216)(11 145 72 213)(12 150 65 210)(13 147 66 215)(14 152 67 212)(15 149 68 209)(16 146 69 214)(17 73 58 217)(18 78 59 222)(19 75 60 219)(20 80 61 224)(21 77 62 221)(22 74 63 218)(23 79 64 223)(24 76 57 220)(25 207 120 129)(26 204 113 134)(27 201 114 131)(28 206 115 136)(29 203 116 133)(30 208 117 130)(31 205 118 135)(32 202 119 132)(33 50 128 141)(34 55 121 138)(35 52 122 143)(36 49 123 140)(37 54 124 137)(38 51 125 142)(39 56 126 139)(40 53 127 144)(81 91 183 199)(82 96 184 196)(83 93 177 193)(84 90 178 198)(85 95 179 195)(86 92 180 200)(87 89 181 197)(88 94 182 194)(97 185 175 112)(98 190 176 109)(99 187 169 106)(100 192 170 111)(101 189 171 108)(102 186 172 105)(103 191 173 110)(104 188 174 107)
(9 70)(10 71)(11 72)(12 65)(13 66)(14 67)(15 68)(16 69)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 57)(49 140)(50 141)(51 142)(52 143)(53 144)(54 137)(55 138)(56 139)(89 197)(90 198)(91 199)(92 200)(93 193)(94 194)(95 195)(96 196)(97 175)(98 176)(99 169)(100 170)(101 171)(102 172)(103 173)(104 174)(129 207)(130 208)(131 201)(132 202)(133 203)(134 204)(135 205)(136 206)(153 164)(154 165)(155 166)(156 167)(157 168)(158 161)(159 162)(160 163)
G:=sub<Sym(224)| (1,219,187,122,119,182,214)(2,215,183,120,123,188,220)(3,221,189,124,113,184,216)(4,209,177,114,125,190,222)(5,223,191,126,115,178,210)(6,211,179,116,127,192,224)(7,217,185,128,117,180,212)(8,213,181,118,121,186,218)(9,154,20,170,144,133,195)(10,196,134,137,171,21,155)(11,156,22,172,138,135,197)(12,198,136,139,173,23,157)(13,158,24,174,140,129,199)(14,200,130,141,175,17,159)(15,160,18,176,142,131,193)(16,194,132,143,169,19,153)(25,36,107,76,48,147,81)(26,82,148,41,77,108,37)(27,38,109,78,42,149,83)(28,84,150,43,79,110,39)(29,40,111,80,44,151,85)(30,86,152,45,73,112,33)(31,34,105,74,46,145,87)(32,88,146,47,75,106,35)(49,207,91,66,161,57,104)(50,97,58,162,67,92,208)(51,201,93,68,163,59,98)(52,99,60,164,69,94,202)(53,203,95,70,165,61,100)(54,101,62,166,71,96,204)(55,205,89,72,167,63,102)(56,103,64,168,65,90,206), (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), (1,153,47,164)(2,158,48,161)(3,155,41,166)(4,160,42,163)(5,157,43,168)(6,154,44,165)(7,159,45,162)(8,156,46,167)(9,151,70,211)(10,148,71,216)(11,145,72,213)(12,150,65,210)(13,147,66,215)(14,152,67,212)(15,149,68,209)(16,146,69,214)(17,73,58,217)(18,78,59,222)(19,75,60,219)(20,80,61,224)(21,77,62,221)(22,74,63,218)(23,79,64,223)(24,76,57,220)(25,207,120,129)(26,204,113,134)(27,201,114,131)(28,206,115,136)(29,203,116,133)(30,208,117,130)(31,205,118,135)(32,202,119,132)(33,50,128,141)(34,55,121,138)(35,52,122,143)(36,49,123,140)(37,54,124,137)(38,51,125,142)(39,56,126,139)(40,53,127,144)(81,91,183,199)(82,96,184,196)(83,93,177,193)(84,90,178,198)(85,95,179,195)(86,92,180,200)(87,89,181,197)(88,94,182,194)(97,185,175,112)(98,190,176,109)(99,187,169,106)(100,192,170,111)(101,189,171,108)(102,186,172,105)(103,191,173,110)(104,188,174,107), (9,70)(10,71)(11,72)(12,65)(13,66)(14,67)(15,68)(16,69)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,57)(49,140)(50,141)(51,142)(52,143)(53,144)(54,137)(55,138)(56,139)(89,197)(90,198)(91,199)(92,200)(93,193)(94,194)(95,195)(96,196)(97,175)(98,176)(99,169)(100,170)(101,171)(102,172)(103,173)(104,174)(129,207)(130,208)(131,201)(132,202)(133,203)(134,204)(135,205)(136,206)(153,164)(154,165)(155,166)(156,167)(157,168)(158,161)(159,162)(160,163)>;
G:=Group( (1,219,187,122,119,182,214)(2,215,183,120,123,188,220)(3,221,189,124,113,184,216)(4,209,177,114,125,190,222)(5,223,191,126,115,178,210)(6,211,179,116,127,192,224)(7,217,185,128,117,180,212)(8,213,181,118,121,186,218)(9,154,20,170,144,133,195)(10,196,134,137,171,21,155)(11,156,22,172,138,135,197)(12,198,136,139,173,23,157)(13,158,24,174,140,129,199)(14,200,130,141,175,17,159)(15,160,18,176,142,131,193)(16,194,132,143,169,19,153)(25,36,107,76,48,147,81)(26,82,148,41,77,108,37)(27,38,109,78,42,149,83)(28,84,150,43,79,110,39)(29,40,111,80,44,151,85)(30,86,152,45,73,112,33)(31,34,105,74,46,145,87)(32,88,146,47,75,106,35)(49,207,91,66,161,57,104)(50,97,58,162,67,92,208)(51,201,93,68,163,59,98)(52,99,60,164,69,94,202)(53,203,95,70,165,61,100)(54,101,62,166,71,96,204)(55,205,89,72,167,63,102)(56,103,64,168,65,90,206), (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), (1,153,47,164)(2,158,48,161)(3,155,41,166)(4,160,42,163)(5,157,43,168)(6,154,44,165)(7,159,45,162)(8,156,46,167)(9,151,70,211)(10,148,71,216)(11,145,72,213)(12,150,65,210)(13,147,66,215)(14,152,67,212)(15,149,68,209)(16,146,69,214)(17,73,58,217)(18,78,59,222)(19,75,60,219)(20,80,61,224)(21,77,62,221)(22,74,63,218)(23,79,64,223)(24,76,57,220)(25,207,120,129)(26,204,113,134)(27,201,114,131)(28,206,115,136)(29,203,116,133)(30,208,117,130)(31,205,118,135)(32,202,119,132)(33,50,128,141)(34,55,121,138)(35,52,122,143)(36,49,123,140)(37,54,124,137)(38,51,125,142)(39,56,126,139)(40,53,127,144)(81,91,183,199)(82,96,184,196)(83,93,177,193)(84,90,178,198)(85,95,179,195)(86,92,180,200)(87,89,181,197)(88,94,182,194)(97,185,175,112)(98,190,176,109)(99,187,169,106)(100,192,170,111)(101,189,171,108)(102,186,172,105)(103,191,173,110)(104,188,174,107), (9,70)(10,71)(11,72)(12,65)(13,66)(14,67)(15,68)(16,69)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,57)(49,140)(50,141)(51,142)(52,143)(53,144)(54,137)(55,138)(56,139)(89,197)(90,198)(91,199)(92,200)(93,193)(94,194)(95,195)(96,196)(97,175)(98,176)(99,169)(100,170)(101,171)(102,172)(103,173)(104,174)(129,207)(130,208)(131,201)(132,202)(133,203)(134,204)(135,205)(136,206)(153,164)(154,165)(155,166)(156,167)(157,168)(158,161)(159,162)(160,163) );
G=PermutationGroup([[(1,219,187,122,119,182,214),(2,215,183,120,123,188,220),(3,221,189,124,113,184,216),(4,209,177,114,125,190,222),(5,223,191,126,115,178,210),(6,211,179,116,127,192,224),(7,217,185,128,117,180,212),(8,213,181,118,121,186,218),(9,154,20,170,144,133,195),(10,196,134,137,171,21,155),(11,156,22,172,138,135,197),(12,198,136,139,173,23,157),(13,158,24,174,140,129,199),(14,200,130,141,175,17,159),(15,160,18,176,142,131,193),(16,194,132,143,169,19,153),(25,36,107,76,48,147,81),(26,82,148,41,77,108,37),(27,38,109,78,42,149,83),(28,84,150,43,79,110,39),(29,40,111,80,44,151,85),(30,86,152,45,73,112,33),(31,34,105,74,46,145,87),(32,88,146,47,75,106,35),(49,207,91,66,161,57,104),(50,97,58,162,67,92,208),(51,201,93,68,163,59,98),(52,99,60,164,69,94,202),(53,203,95,70,165,61,100),(54,101,62,166,71,96,204),(55,205,89,72,167,63,102),(56,103,64,168,65,90,206)], [(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)], [(1,153,47,164),(2,158,48,161),(3,155,41,166),(4,160,42,163),(5,157,43,168),(6,154,44,165),(7,159,45,162),(8,156,46,167),(9,151,70,211),(10,148,71,216),(11,145,72,213),(12,150,65,210),(13,147,66,215),(14,152,67,212),(15,149,68,209),(16,146,69,214),(17,73,58,217),(18,78,59,222),(19,75,60,219),(20,80,61,224),(21,77,62,221),(22,74,63,218),(23,79,64,223),(24,76,57,220),(25,207,120,129),(26,204,113,134),(27,201,114,131),(28,206,115,136),(29,203,116,133),(30,208,117,130),(31,205,118,135),(32,202,119,132),(33,50,128,141),(34,55,121,138),(35,52,122,143),(36,49,123,140),(37,54,124,137),(38,51,125,142),(39,56,126,139),(40,53,127,144),(81,91,183,199),(82,96,184,196),(83,93,177,193),(84,90,178,198),(85,95,179,195),(86,92,180,200),(87,89,181,197),(88,94,182,194),(97,185,175,112),(98,190,176,109),(99,187,169,106),(100,192,170,111),(101,189,171,108),(102,186,172,105),(103,191,173,110),(104,188,174,107)], [(9,70),(10,71),(11,72),(12,65),(13,66),(14,67),(15,68),(16,69),(17,58),(18,59),(19,60),(20,61),(21,62),(22,63),(23,64),(24,57),(49,140),(50,141),(51,142),(52,143),(53,144),(54,137),(55,138),(56,139),(89,197),(90,198),(91,199),(92,200),(93,193),(94,194),(95,195),(96,196),(97,175),(98,176),(99,169),(100,170),(101,171),(102,172),(103,173),(104,174),(129,207),(130,208),(131,201),(132,202),(133,203),(134,204),(135,205),(136,206),(153,164),(154,165),(155,166),(156,167),(157,168),(158,161),(159,162),(160,163)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D7 | C4○D4 | M4(2) | D14 | D14 | C8○D4 | C4×D7 | C4×D7 | C8⋊D7 | D4×D7 | D4⋊2D7 | D28.C4 |
| kernel | C7⋊C8⋊26D4 | Dic7⋊C8 | C56⋊C4 | D14⋊C8 | C7×C22⋊C8 | C2×C8⋊D7 | C22×C7⋊C8 | C4×C7⋊D4 | Dic7⋊C4 | D14⋊C4 | C23.D7 | C2×C7⋊D4 | C7⋊C8 | C22⋊C8 | C28 | C2×C14 | C2×C8 | C22×C4 | C14 | C2×C4 | C23 | C22 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 2 | 4 | 6 | 3 | 4 | 6 | 6 | 24 | 3 | 3 | 6 |
Matrix representation of C7⋊C8⋊26D4 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 34 | 112 |
| 0 | 0 | 60 | 88 |
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 47 | 95 |
| 0 | 0 | 111 | 66 |
| 0 | 15 | 0 | 0 |
| 15 | 0 | 0 | 0 |
| 0 | 0 | 13 | 55 |
| 0 | 0 | 112 | 100 |
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,34,60,0,0,112,88],[15,0,0,0,0,15,0,0,0,0,47,111,0,0,95,66],[0,15,0,0,15,0,0,0,0,0,13,112,0,0,55,100],[1,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1] >;
C7⋊C8⋊26D4 in GAP, Magma, Sage, TeX
C_7\rtimes C_8\rtimes_{26}D_4 % in TeX
G:=Group("C7:C8:26D4"); // GroupNames label
G:=SmallGroup(448,264);
// by ID
G=gap.SmallGroup(448,264);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,758,219,58,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=c^4=d^2=1,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=b^5,b*d=d*b,d*c*d=c^-1>;
// generators/relations