direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8×D31, C4.6D62, Dic62⋊4C2, C62.7C23, C124.6C22, D62.5C22, Dic31.3C22, C31⋊2(C2×Q8), (Q8×C31)⋊2C2, (C4×D31).1C2, C2.8(C22×D31), SmallGroup(496,33)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×D31
G = < a,b,c,d | a4=c31=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of Q8×D31
►On 248 points
Generators in S248
(1 108 48 73)(2 109 49 74)(3 110 50 75)(4 111 51 76)(5 112 52 77)(6 113 53 78)(7 114 54 79)(8 115 55 80)(9 116 56 81)(10 117 57 82)(11 118 58 83)(12 119 59 84)(13 120 60 85)(14 121 61 86)(15 122 62 87)(16 123 32 88)(17 124 33 89)(18 94 34 90)(19 95 35 91)(20 96 36 92)(21 97 37 93)(22 98 38 63)(23 99 39 64)(24 100 40 65)(25 101 41 66)(26 102 42 67)(27 103 43 68)(28 104 44 69)(29 105 45 70)(30 106 46 71)(31 107 47 72)(125 208 172 244)(126 209 173 245)(127 210 174 246)(128 211 175 247)(129 212 176 248)(130 213 177 218)(131 214 178 219)(132 215 179 220)(133 216 180 221)(134 217 181 222)(135 187 182 223)(136 188 183 224)(137 189 184 225)(138 190 185 226)(139 191 186 227)(140 192 156 228)(141 193 157 229)(142 194 158 230)(143 195 159 231)(144 196 160 232)(145 197 161 233)(146 198 162 234)(147 199 163 235)(148 200 164 236)(149 201 165 237)(150 202 166 238)(151 203 167 239)(152 204 168 240)(153 205 169 241)(154 206 170 242)(155 207 171 243)
(1 165 48 149)(2 166 49 150)(3 167 50 151)(4 168 51 152)(5 169 52 153)(6 170 53 154)(7 171 54 155)(8 172 55 125)(9 173 56 126)(10 174 57 127)(11 175 58 128)(12 176 59 129)(13 177 60 130)(14 178 61 131)(15 179 62 132)(16 180 32 133)(17 181 33 134)(18 182 34 135)(19 183 35 136)(20 184 36 137)(21 185 37 138)(22 186 38 139)(23 156 39 140)(24 157 40 141)(25 158 41 142)(26 159 42 143)(27 160 43 144)(28 161 44 145)(29 162 45 146)(30 163 46 147)(31 164 47 148)(63 227 98 191)(64 228 99 192)(65 229 100 193)(66 230 101 194)(67 231 102 195)(68 232 103 196)(69 233 104 197)(70 234 105 198)(71 235 106 199)(72 236 107 200)(73 237 108 201)(74 238 109 202)(75 239 110 203)(76 240 111 204)(77 241 112 205)(78 242 113 206)(79 243 114 207)(80 244 115 208)(81 245 116 209)(82 246 117 210)(83 247 118 211)(84 248 119 212)(85 218 120 213)(86 219 121 214)(87 220 122 215)(88 221 123 216)(89 222 124 217)(90 223 94 187)(91 224 95 188)(92 225 96 189)(93 226 97 190)
(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)(218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248)
(1 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 21)(12 20)(13 19)(14 18)(15 17)(33 62)(34 61)(35 60)(36 59)(37 58)(38 57)(39 56)(40 55)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 48)(63 82)(64 81)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)(83 93)(84 92)(85 91)(86 90)(87 89)(94 121)(95 120)(96 119)(97 118)(98 117)(99 116)(100 115)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)(122 124)(125 141)(126 140)(127 139)(128 138)(129 137)(130 136)(131 135)(132 134)(142 155)(143 154)(144 153)(145 152)(146 151)(147 150)(148 149)(156 173)(157 172)(158 171)(159 170)(160 169)(161 168)(162 167)(163 166)(164 165)(174 186)(175 185)(176 184)(177 183)(178 182)(179 181)(187 214)(188 213)(189 212)(190 211)(191 210)(192 209)(193 208)(194 207)(195 206)(196 205)(197 204)(198 203)(199 202)(200 201)(215 217)(218 224)(219 223)(220 222)(225 248)(226 247)(227 246)(228 245)(229 244)(230 243)(231 242)(232 241)(233 240)(234 239)(235 238)(236 237)
G:=sub<Sym(248)| (1,108,48,73)(2,109,49,74)(3,110,50,75)(4,111,51,76)(5,112,52,77)(6,113,53,78)(7,114,54,79)(8,115,55,80)(9,116,56,81)(10,117,57,82)(11,118,58,83)(12,119,59,84)(13,120,60,85)(14,121,61,86)(15,122,62,87)(16,123,32,88)(17,124,33,89)(18,94,34,90)(19,95,35,91)(20,96,36,92)(21,97,37,93)(22,98,38,63)(23,99,39,64)(24,100,40,65)(25,101,41,66)(26,102,42,67)(27,103,43,68)(28,104,44,69)(29,105,45,70)(30,106,46,71)(31,107,47,72)(125,208,172,244)(126,209,173,245)(127,210,174,246)(128,211,175,247)(129,212,176,248)(130,213,177,218)(131,214,178,219)(132,215,179,220)(133,216,180,221)(134,217,181,222)(135,187,182,223)(136,188,183,224)(137,189,184,225)(138,190,185,226)(139,191,186,227)(140,192,156,228)(141,193,157,229)(142,194,158,230)(143,195,159,231)(144,196,160,232)(145,197,161,233)(146,198,162,234)(147,199,163,235)(148,200,164,236)(149,201,165,237)(150,202,166,238)(151,203,167,239)(152,204,168,240)(153,205,169,241)(154,206,170,242)(155,207,171,243), (1,165,48,149)(2,166,49,150)(3,167,50,151)(4,168,51,152)(5,169,52,153)(6,170,53,154)(7,171,54,155)(8,172,55,125)(9,173,56,126)(10,174,57,127)(11,175,58,128)(12,176,59,129)(13,177,60,130)(14,178,61,131)(15,179,62,132)(16,180,32,133)(17,181,33,134)(18,182,34,135)(19,183,35,136)(20,184,36,137)(21,185,37,138)(22,186,38,139)(23,156,39,140)(24,157,40,141)(25,158,41,142)(26,159,42,143)(27,160,43,144)(28,161,44,145)(29,162,45,146)(30,163,46,147)(31,164,47,148)(63,227,98,191)(64,228,99,192)(65,229,100,193)(66,230,101,194)(67,231,102,195)(68,232,103,196)(69,233,104,197)(70,234,105,198)(71,235,106,199)(72,236,107,200)(73,237,108,201)(74,238,109,202)(75,239,110,203)(76,240,111,204)(77,241,112,205)(78,242,113,206)(79,243,114,207)(80,244,115,208)(81,245,116,209)(82,246,117,210)(83,247,118,211)(84,248,119,212)(85,218,120,213)(86,219,121,214)(87,220,122,215)(88,221,123,216)(89,222,124,217)(90,223,94,187)(91,224,95,188)(92,225,96,189)(93,226,97,190), (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)(218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(33,62)(34,61)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(83,93)(84,92)(85,91)(86,90)(87,89)(94,121)(95,120)(96,119)(97,118)(98,117)(99,116)(100,115)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)(122,124)(125,141)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(142,155)(143,154)(144,153)(145,152)(146,151)(147,150)(148,149)(156,173)(157,172)(158,171)(159,170)(160,169)(161,168)(162,167)(163,166)(164,165)(174,186)(175,185)(176,184)(177,183)(178,182)(179,181)(187,214)(188,213)(189,212)(190,211)(191,210)(192,209)(193,208)(194,207)(195,206)(196,205)(197,204)(198,203)(199,202)(200,201)(215,217)(218,224)(219,223)(220,222)(225,248)(226,247)(227,246)(228,245)(229,244)(230,243)(231,242)(232,241)(233,240)(234,239)(235,238)(236,237)>;
G:=Group( (1,108,48,73)(2,109,49,74)(3,110,50,75)(4,111,51,76)(5,112,52,77)(6,113,53,78)(7,114,54,79)(8,115,55,80)(9,116,56,81)(10,117,57,82)(11,118,58,83)(12,119,59,84)(13,120,60,85)(14,121,61,86)(15,122,62,87)(16,123,32,88)(17,124,33,89)(18,94,34,90)(19,95,35,91)(20,96,36,92)(21,97,37,93)(22,98,38,63)(23,99,39,64)(24,100,40,65)(25,101,41,66)(26,102,42,67)(27,103,43,68)(28,104,44,69)(29,105,45,70)(30,106,46,71)(31,107,47,72)(125,208,172,244)(126,209,173,245)(127,210,174,246)(128,211,175,247)(129,212,176,248)(130,213,177,218)(131,214,178,219)(132,215,179,220)(133,216,180,221)(134,217,181,222)(135,187,182,223)(136,188,183,224)(137,189,184,225)(138,190,185,226)(139,191,186,227)(140,192,156,228)(141,193,157,229)(142,194,158,230)(143,195,159,231)(144,196,160,232)(145,197,161,233)(146,198,162,234)(147,199,163,235)(148,200,164,236)(149,201,165,237)(150,202,166,238)(151,203,167,239)(152,204,168,240)(153,205,169,241)(154,206,170,242)(155,207,171,243), (1,165,48,149)(2,166,49,150)(3,167,50,151)(4,168,51,152)(5,169,52,153)(6,170,53,154)(7,171,54,155)(8,172,55,125)(9,173,56,126)(10,174,57,127)(11,175,58,128)(12,176,59,129)(13,177,60,130)(14,178,61,131)(15,179,62,132)(16,180,32,133)(17,181,33,134)(18,182,34,135)(19,183,35,136)(20,184,36,137)(21,185,37,138)(22,186,38,139)(23,156,39,140)(24,157,40,141)(25,158,41,142)(26,159,42,143)(27,160,43,144)(28,161,44,145)(29,162,45,146)(30,163,46,147)(31,164,47,148)(63,227,98,191)(64,228,99,192)(65,229,100,193)(66,230,101,194)(67,231,102,195)(68,232,103,196)(69,233,104,197)(70,234,105,198)(71,235,106,199)(72,236,107,200)(73,237,108,201)(74,238,109,202)(75,239,110,203)(76,240,111,204)(77,241,112,205)(78,242,113,206)(79,243,114,207)(80,244,115,208)(81,245,116,209)(82,246,117,210)(83,247,118,211)(84,248,119,212)(85,218,120,213)(86,219,121,214)(87,220,122,215)(88,221,123,216)(89,222,124,217)(90,223,94,187)(91,224,95,188)(92,225,96,189)(93,226,97,190), (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)(218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(33,62)(34,61)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(83,93)(84,92)(85,91)(86,90)(87,89)(94,121)(95,120)(96,119)(97,118)(98,117)(99,116)(100,115)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)(122,124)(125,141)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(142,155)(143,154)(144,153)(145,152)(146,151)(147,150)(148,149)(156,173)(157,172)(158,171)(159,170)(160,169)(161,168)(162,167)(163,166)(164,165)(174,186)(175,185)(176,184)(177,183)(178,182)(179,181)(187,214)(188,213)(189,212)(190,211)(191,210)(192,209)(193,208)(194,207)(195,206)(196,205)(197,204)(198,203)(199,202)(200,201)(215,217)(218,224)(219,223)(220,222)(225,248)(226,247)(227,246)(228,245)(229,244)(230,243)(231,242)(232,241)(233,240)(234,239)(235,238)(236,237) );
G=PermutationGroup([[(1,108,48,73),(2,109,49,74),(3,110,50,75),(4,111,51,76),(5,112,52,77),(6,113,53,78),(7,114,54,79),(8,115,55,80),(9,116,56,81),(10,117,57,82),(11,118,58,83),(12,119,59,84),(13,120,60,85),(14,121,61,86),(15,122,62,87),(16,123,32,88),(17,124,33,89),(18,94,34,90),(19,95,35,91),(20,96,36,92),(21,97,37,93),(22,98,38,63),(23,99,39,64),(24,100,40,65),(25,101,41,66),(26,102,42,67),(27,103,43,68),(28,104,44,69),(29,105,45,70),(30,106,46,71),(31,107,47,72),(125,208,172,244),(126,209,173,245),(127,210,174,246),(128,211,175,247),(129,212,176,248),(130,213,177,218),(131,214,178,219),(132,215,179,220),(133,216,180,221),(134,217,181,222),(135,187,182,223),(136,188,183,224),(137,189,184,225),(138,190,185,226),(139,191,186,227),(140,192,156,228),(141,193,157,229),(142,194,158,230),(143,195,159,231),(144,196,160,232),(145,197,161,233),(146,198,162,234),(147,199,163,235),(148,200,164,236),(149,201,165,237),(150,202,166,238),(151,203,167,239),(152,204,168,240),(153,205,169,241),(154,206,170,242),(155,207,171,243)], [(1,165,48,149),(2,166,49,150),(3,167,50,151),(4,168,51,152),(5,169,52,153),(6,170,53,154),(7,171,54,155),(8,172,55,125),(9,173,56,126),(10,174,57,127),(11,175,58,128),(12,176,59,129),(13,177,60,130),(14,178,61,131),(15,179,62,132),(16,180,32,133),(17,181,33,134),(18,182,34,135),(19,183,35,136),(20,184,36,137),(21,185,37,138),(22,186,38,139),(23,156,39,140),(24,157,40,141),(25,158,41,142),(26,159,42,143),(27,160,43,144),(28,161,44,145),(29,162,45,146),(30,163,46,147),(31,164,47,148),(63,227,98,191),(64,228,99,192),(65,229,100,193),(66,230,101,194),(67,231,102,195),(68,232,103,196),(69,233,104,197),(70,234,105,198),(71,235,106,199),(72,236,107,200),(73,237,108,201),(74,238,109,202),(75,239,110,203),(76,240,111,204),(77,241,112,205),(78,242,113,206),(79,243,114,207),(80,244,115,208),(81,245,116,209),(82,246,117,210),(83,247,118,211),(84,248,119,212),(85,218,120,213),(86,219,121,214),(87,220,122,215),(88,221,123,216),(89,222,124,217),(90,223,94,187),(91,224,95,188),(92,225,96,189),(93,226,97,190)], [(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),(218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248)], [(1,31),(2,30),(3,29),(4,28),(5,27),(6,26),(7,25),(8,24),(9,23),(10,22),(11,21),(12,20),(13,19),(14,18),(15,17),(33,62),(34,61),(35,60),(36,59),(37,58),(38,57),(39,56),(40,55),(41,54),(42,53),(43,52),(44,51),(45,50),(46,49),(47,48),(63,82),(64,81),(65,80),(66,79),(67,78),(68,77),(69,76),(70,75),(71,74),(72,73),(83,93),(84,92),(85,91),(86,90),(87,89),(94,121),(95,120),(96,119),(97,118),(98,117),(99,116),(100,115),(101,114),(102,113),(103,112),(104,111),(105,110),(106,109),(107,108),(122,124),(125,141),(126,140),(127,139),(128,138),(129,137),(130,136),(131,135),(132,134),(142,155),(143,154),(144,153),(145,152),(146,151),(147,150),(148,149),(156,173),(157,172),(158,171),(159,170),(160,169),(161,168),(162,167),(163,166),(164,165),(174,186),(175,185),(176,184),(177,183),(178,182),(179,181),(187,214),(188,213),(189,212),(190,211),(191,210),(192,209),(193,208),(194,207),(195,206),(196,205),(197,204),(198,203),(199,202),(200,201),(215,217),(218,224),(219,223),(220,222),(225,248),(226,247),(227,246),(228,245),(229,244),(230,243),(231,242),(232,241),(233,240),(234,239),(235,238),(236,237)]])
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 31A | ··· | 31O | 62A | ··· | 62O | 124A | ··· | 124AS |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 |
| size | 1 | 1 | 31 | 31 | 2 | 2 | 2 | 62 | 62 | 62 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | Q8 | D31 | D62 | Q8×D31 |
| kernel | Q8×D31 | Dic62 | C4×D31 | Q8×C31 | D31 | Q8 | C4 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 15 | 45 | 15 |
Matrix representation of Q8×D31 ►in GL4(𝔽373) generated by
| 372 | 181 | 0 | 0 |
| 136 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 36 | 229 | 0 | 0 |
| 224 | 337 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 345 | 1 |
| 0 | 0 | 124 | 222 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 201 | 194 |
| 0 | 0 | 84 | 172 |
G:=sub<GL(4,GF(373))| [372,136,0,0,181,1,0,0,0,0,1,0,0,0,0,1],[36,224,0,0,229,337,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,345,124,0,0,1,222],[1,0,0,0,0,1,0,0,0,0,201,84,0,0,194,172] >;
Q8×D31 in GAP, Magma, Sage, TeX
Q_8\times D_{31} % in TeX
G:=Group("Q8xD31"); // GroupNames label
G:=SmallGroup(496,33);
// by ID
G=gap.SmallGroup(496,33);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-31,46,97,42,12004]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^31=d^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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