metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q32⋊3D7, D112⋊5C2, D14.3D8, C16.10D14, Q16.5D14, C112.8C22, C56.22C23, Dic7.14D8, D56.5C22, C7⋊C8.16D4, (D7×C16)⋊3C2, C7⋊4(C4○D16), (C7×Q32)⋊3C2, C4.10(D4×D7), C2.25(D7×D8), (C4×D7).23D4, C28.16(C2×D4), C14.41(C2×D8), C7⋊SD32⋊4C2, C7⋊C16.8C22, Q8.D14⋊5C2, C8.28(C22×D7), (C8×D7).14C22, (C7×Q16).6C22, SmallGroup(448,453)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q32⋊3D7
G = < a,b,c,d | a16=c7=d2=1, b2=a8, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd=a8b, dcd=c-1 >
Subgroups: 592 in 84 conjugacy classes, 31 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, D4, Q8, D7, C14, C16, C16, C2×C8, D8, SD16, Q16, C4○D4, Dic7, C28, C28, D14, D14, C2×C16, D16, SD32, Q32, C4○D8, C7⋊C8, C56, C4×D7, C4×D7, D28, C7×Q8, C4○D16, C7⋊C16, C112, C8×D7, D56, Q8⋊D7, C7×Q16, Q8⋊2D7, D7×C16, D112, C7⋊SD32, C7×Q32, Q8.D14, Q32⋊3D7
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C2×D8, C22×D7, C4○D16, D4×D7, D7×D8, Q32⋊3D7
►On 224 points
Generators in S224
(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)
(1 149 9 157)(2 148 10 156)(3 147 11 155)(4 146 12 154)(5 145 13 153)(6 160 14 152)(7 159 15 151)(8 158 16 150)(17 179 25 187)(18 178 26 186)(19 177 27 185)(20 192 28 184)(21 191 29 183)(22 190 30 182)(23 189 31 181)(24 188 32 180)(33 77 41 69)(34 76 42 68)(35 75 43 67)(36 74 44 66)(37 73 45 65)(38 72 46 80)(39 71 47 79)(40 70 48 78)(49 129 57 137)(50 144 58 136)(51 143 59 135)(52 142 60 134)(53 141 61 133)(54 140 62 132)(55 139 63 131)(56 138 64 130)(81 105 89 97)(82 104 90 112)(83 103 91 111)(84 102 92 110)(85 101 93 109)(86 100 94 108)(87 99 95 107)(88 98 96 106)(113 223 121 215)(114 222 122 214)(115 221 123 213)(116 220 124 212)(117 219 125 211)(118 218 126 210)(119 217 127 209)(120 216 128 224)(161 202 169 194)(162 201 170 193)(163 200 171 208)(164 199 172 207)(165 198 173 206)(166 197 174 205)(167 196 175 204)(168 195 176 203)
(1 190 106 125 58 40 208)(2 191 107 126 59 41 193)(3 192 108 127 60 42 194)(4 177 109 128 61 43 195)(5 178 110 113 62 44 196)(6 179 111 114 63 45 197)(7 180 112 115 64 46 198)(8 181 97 116 49 47 199)(9 182 98 117 50 48 200)(10 183 99 118 51 33 201)(11 184 100 119 52 34 202)(12 185 101 120 53 35 203)(13 186 102 121 54 36 204)(14 187 103 122 55 37 205)(15 188 104 123 56 38 206)(16 189 105 124 57 39 207)(17 91 214 139 73 166 152)(18 92 215 140 74 167 153)(19 93 216 141 75 168 154)(20 94 217 142 76 169 155)(21 95 218 143 77 170 156)(22 96 219 144 78 171 157)(23 81 220 129 79 172 158)(24 82 221 130 80 173 159)(25 83 222 131 65 174 160)(26 84 223 132 66 175 145)(27 85 224 133 67 176 146)(28 86 209 134 68 161 147)(29 87 210 135 69 162 148)(30 88 211 136 70 163 149)(31 89 212 137 71 164 150)(32 90 213 138 72 165 151)
(1 208)(2 193)(3 194)(4 195)(5 196)(6 197)(7 198)(8 199)(9 200)(10 201)(11 202)(12 203)(13 204)(14 205)(15 206)(16 207)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 183)(34 184)(35 185)(36 186)(37 187)(38 188)(39 189)(40 190)(41 191)(42 192)(43 177)(44 178)(45 179)(46 180)(47 181)(48 182)(49 97)(50 98)(51 99)(52 100)(53 101)(54 102)(55 103)(56 104)(57 105)(58 106)(59 107)(60 108)(61 109)(62 110)(63 111)(64 112)(81 137)(82 138)(83 139)(84 140)(85 141)(86 142)(87 143)(88 144)(89 129)(90 130)(91 131)(92 132)(93 133)(94 134)(95 135)(96 136)(145 167)(146 168)(147 169)(148 170)(149 171)(150 172)(151 173)(152 174)(153 175)(154 176)(155 161)(156 162)(157 163)(158 164)(159 165)(160 166)(209 217)(210 218)(211 219)(212 220)(213 221)(214 222)(215 223)(216 224)
G:=sub<Sym(224)| (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), (1,149,9,157)(2,148,10,156)(3,147,11,155)(4,146,12,154)(5,145,13,153)(6,160,14,152)(7,159,15,151)(8,158,16,150)(17,179,25,187)(18,178,26,186)(19,177,27,185)(20,192,28,184)(21,191,29,183)(22,190,30,182)(23,189,31,181)(24,188,32,180)(33,77,41,69)(34,76,42,68)(35,75,43,67)(36,74,44,66)(37,73,45,65)(38,72,46,80)(39,71,47,79)(40,70,48,78)(49,129,57,137)(50,144,58,136)(51,143,59,135)(52,142,60,134)(53,141,61,133)(54,140,62,132)(55,139,63,131)(56,138,64,130)(81,105,89,97)(82,104,90,112)(83,103,91,111)(84,102,92,110)(85,101,93,109)(86,100,94,108)(87,99,95,107)(88,98,96,106)(113,223,121,215)(114,222,122,214)(115,221,123,213)(116,220,124,212)(117,219,125,211)(118,218,126,210)(119,217,127,209)(120,216,128,224)(161,202,169,194)(162,201,170,193)(163,200,171,208)(164,199,172,207)(165,198,173,206)(166,197,174,205)(167,196,175,204)(168,195,176,203), (1,190,106,125,58,40,208)(2,191,107,126,59,41,193)(3,192,108,127,60,42,194)(4,177,109,128,61,43,195)(5,178,110,113,62,44,196)(6,179,111,114,63,45,197)(7,180,112,115,64,46,198)(8,181,97,116,49,47,199)(9,182,98,117,50,48,200)(10,183,99,118,51,33,201)(11,184,100,119,52,34,202)(12,185,101,120,53,35,203)(13,186,102,121,54,36,204)(14,187,103,122,55,37,205)(15,188,104,123,56,38,206)(16,189,105,124,57,39,207)(17,91,214,139,73,166,152)(18,92,215,140,74,167,153)(19,93,216,141,75,168,154)(20,94,217,142,76,169,155)(21,95,218,143,77,170,156)(22,96,219,144,78,171,157)(23,81,220,129,79,172,158)(24,82,221,130,80,173,159)(25,83,222,131,65,174,160)(26,84,223,132,66,175,145)(27,85,224,133,67,176,146)(28,86,209,134,68,161,147)(29,87,210,135,69,162,148)(30,88,211,136,70,163,149)(31,89,212,137,71,164,150)(32,90,213,138,72,165,151), (1,208)(2,193)(3,194)(4,195)(5,196)(6,197)(7,198)(8,199)(9,200)(10,201)(11,202)(12,203)(13,204)(14,205)(15,206)(16,207)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,183)(34,184)(35,185)(36,186)(37,187)(38,188)(39,189)(40,190)(41,191)(42,192)(43,177)(44,178)(45,179)(46,180)(47,181)(48,182)(49,97)(50,98)(51,99)(52,100)(53,101)(54,102)(55,103)(56,104)(57,105)(58,106)(59,107)(60,108)(61,109)(62,110)(63,111)(64,112)(81,137)(82,138)(83,139)(84,140)(85,141)(86,142)(87,143)(88,144)(89,129)(90,130)(91,131)(92,132)(93,133)(94,134)(95,135)(96,136)(145,167)(146,168)(147,169)(148,170)(149,171)(150,172)(151,173)(152,174)(153,175)(154,176)(155,161)(156,162)(157,163)(158,164)(159,165)(160,166)(209,217)(210,218)(211,219)(212,220)(213,221)(214,222)(215,223)(216,224)>;
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), (1,149,9,157)(2,148,10,156)(3,147,11,155)(4,146,12,154)(5,145,13,153)(6,160,14,152)(7,159,15,151)(8,158,16,150)(17,179,25,187)(18,178,26,186)(19,177,27,185)(20,192,28,184)(21,191,29,183)(22,190,30,182)(23,189,31,181)(24,188,32,180)(33,77,41,69)(34,76,42,68)(35,75,43,67)(36,74,44,66)(37,73,45,65)(38,72,46,80)(39,71,47,79)(40,70,48,78)(49,129,57,137)(50,144,58,136)(51,143,59,135)(52,142,60,134)(53,141,61,133)(54,140,62,132)(55,139,63,131)(56,138,64,130)(81,105,89,97)(82,104,90,112)(83,103,91,111)(84,102,92,110)(85,101,93,109)(86,100,94,108)(87,99,95,107)(88,98,96,106)(113,223,121,215)(114,222,122,214)(115,221,123,213)(116,220,124,212)(117,219,125,211)(118,218,126,210)(119,217,127,209)(120,216,128,224)(161,202,169,194)(162,201,170,193)(163,200,171,208)(164,199,172,207)(165,198,173,206)(166,197,174,205)(167,196,175,204)(168,195,176,203), (1,190,106,125,58,40,208)(2,191,107,126,59,41,193)(3,192,108,127,60,42,194)(4,177,109,128,61,43,195)(5,178,110,113,62,44,196)(6,179,111,114,63,45,197)(7,180,112,115,64,46,198)(8,181,97,116,49,47,199)(9,182,98,117,50,48,200)(10,183,99,118,51,33,201)(11,184,100,119,52,34,202)(12,185,101,120,53,35,203)(13,186,102,121,54,36,204)(14,187,103,122,55,37,205)(15,188,104,123,56,38,206)(16,189,105,124,57,39,207)(17,91,214,139,73,166,152)(18,92,215,140,74,167,153)(19,93,216,141,75,168,154)(20,94,217,142,76,169,155)(21,95,218,143,77,170,156)(22,96,219,144,78,171,157)(23,81,220,129,79,172,158)(24,82,221,130,80,173,159)(25,83,222,131,65,174,160)(26,84,223,132,66,175,145)(27,85,224,133,67,176,146)(28,86,209,134,68,161,147)(29,87,210,135,69,162,148)(30,88,211,136,70,163,149)(31,89,212,137,71,164,150)(32,90,213,138,72,165,151), (1,208)(2,193)(3,194)(4,195)(5,196)(6,197)(7,198)(8,199)(9,200)(10,201)(11,202)(12,203)(13,204)(14,205)(15,206)(16,207)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,183)(34,184)(35,185)(36,186)(37,187)(38,188)(39,189)(40,190)(41,191)(42,192)(43,177)(44,178)(45,179)(46,180)(47,181)(48,182)(49,97)(50,98)(51,99)(52,100)(53,101)(54,102)(55,103)(56,104)(57,105)(58,106)(59,107)(60,108)(61,109)(62,110)(63,111)(64,112)(81,137)(82,138)(83,139)(84,140)(85,141)(86,142)(87,143)(88,144)(89,129)(90,130)(91,131)(92,132)(93,133)(94,134)(95,135)(96,136)(145,167)(146,168)(147,169)(148,170)(149,171)(150,172)(151,173)(152,174)(153,175)(154,176)(155,161)(156,162)(157,163)(158,164)(159,165)(160,166)(209,217)(210,218)(211,219)(212,220)(213,221)(214,222)(215,223)(216,224) );
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)], [(1,149,9,157),(2,148,10,156),(3,147,11,155),(4,146,12,154),(5,145,13,153),(6,160,14,152),(7,159,15,151),(8,158,16,150),(17,179,25,187),(18,178,26,186),(19,177,27,185),(20,192,28,184),(21,191,29,183),(22,190,30,182),(23,189,31,181),(24,188,32,180),(33,77,41,69),(34,76,42,68),(35,75,43,67),(36,74,44,66),(37,73,45,65),(38,72,46,80),(39,71,47,79),(40,70,48,78),(49,129,57,137),(50,144,58,136),(51,143,59,135),(52,142,60,134),(53,141,61,133),(54,140,62,132),(55,139,63,131),(56,138,64,130),(81,105,89,97),(82,104,90,112),(83,103,91,111),(84,102,92,110),(85,101,93,109),(86,100,94,108),(87,99,95,107),(88,98,96,106),(113,223,121,215),(114,222,122,214),(115,221,123,213),(116,220,124,212),(117,219,125,211),(118,218,126,210),(119,217,127,209),(120,216,128,224),(161,202,169,194),(162,201,170,193),(163,200,171,208),(164,199,172,207),(165,198,173,206),(166,197,174,205),(167,196,175,204),(168,195,176,203)], [(1,190,106,125,58,40,208),(2,191,107,126,59,41,193),(3,192,108,127,60,42,194),(4,177,109,128,61,43,195),(5,178,110,113,62,44,196),(6,179,111,114,63,45,197),(7,180,112,115,64,46,198),(8,181,97,116,49,47,199),(9,182,98,117,50,48,200),(10,183,99,118,51,33,201),(11,184,100,119,52,34,202),(12,185,101,120,53,35,203),(13,186,102,121,54,36,204),(14,187,103,122,55,37,205),(15,188,104,123,56,38,206),(16,189,105,124,57,39,207),(17,91,214,139,73,166,152),(18,92,215,140,74,167,153),(19,93,216,141,75,168,154),(20,94,217,142,76,169,155),(21,95,218,143,77,170,156),(22,96,219,144,78,171,157),(23,81,220,129,79,172,158),(24,82,221,130,80,173,159),(25,83,222,131,65,174,160),(26,84,223,132,66,175,145),(27,85,224,133,67,176,146),(28,86,209,134,68,161,147),(29,87,210,135,69,162,148),(30,88,211,136,70,163,149),(31,89,212,137,71,164,150),(32,90,213,138,72,165,151)], [(1,208),(2,193),(3,194),(4,195),(5,196),(6,197),(7,198),(8,199),(9,200),(10,201),(11,202),(12,203),(13,204),(14,205),(15,206),(16,207),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,183),(34,184),(35,185),(36,186),(37,187),(38,188),(39,189),(40,190),(41,191),(42,192),(43,177),(44,178),(45,179),(46,180),(47,181),(48,182),(49,97),(50,98),(51,99),(52,100),(53,101),(54,102),(55,103),(56,104),(57,105),(58,106),(59,107),(60,108),(61,109),(62,110),(63,111),(64,112),(81,137),(82,138),(83,139),(84,140),(85,141),(86,142),(87,143),(88,144),(89,129),(90,130),(91,131),(92,132),(93,133),(94,134),(95,135),(96,136),(145,167),(146,168),(147,169),(148,170),(149,171),(150,172),(151,173),(152,174),(153,175),(154,176),(155,161),(156,162),(157,163),(158,164),(159,165),(160,166),(209,217),(210,218),(211,219),(212,220),(213,221),(214,222),(215,223),(216,224)]])
55 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 28A | 28B | 28C | 28D | ··· | 28I | 56A | ··· | 56F | 112A | ··· | 112L |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 28 | 28 | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 14 | 56 | 56 | 2 | 7 | 7 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 4 | 4 | 4 | 16 | ··· | 16 | 4 | ··· | 4 | 4 | ··· | 4 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D8 | D8 | D14 | D14 | C4○D16 | D4×D7 | D7×D8 | Q32⋊3D7 |
| kernel | Q32⋊3D7 | D7×C16 | D112 | C7⋊SD32 | C7×Q32 | Q8.D14 | C7⋊C8 | C4×D7 | Q32 | Dic7 | D14 | C16 | Q16 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 8 | 3 | 6 | 12 |
Matrix representation of Q32⋊3D7 ►in GL4(𝔽113) generated by
| 73 | 0 | 0 | 0 |
| 51 | 48 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 106 | 112 | 0 | 0 |
| 50 | 7 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 103 | 1 |
| 0 | 0 | 111 | 34 |
| 1 | 0 | 0 | 0 |
| 99 | 112 | 0 | 0 |
| 0 | 0 | 89 | 24 |
| 0 | 0 | 9 | 24 |
G:=sub<GL(4,GF(113))| [73,51,0,0,0,48,0,0,0,0,112,0,0,0,0,112],[106,50,0,0,112,7,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,103,111,0,0,1,34],[1,99,0,0,0,112,0,0,0,0,89,9,0,0,24,24] >;
Q32⋊3D7 in GAP, Magma, Sage, TeX
Q_{32}\rtimes_3D_7 % in TeX
G:=Group("Q32:3D7"); // GroupNames label
G:=SmallGroup(448,453);
// by ID
G=gap.SmallGroup(448,453);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,232,758,135,184,346,185,192,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^16=c^7=d^2=1,b^2=a^8,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^8*b,d*c*d=c^-1>;
// generators/relations