metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊D31, C62.9D4, C4.3D62, C31⋊3SD16, D124.2C2, C124.3C22, C31⋊C8⋊3C2, (Q8×C31)⋊1C2, C2.6(C31⋊D4), SmallGroup(496,16)
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=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=c-1 >
Smallest permutation representation of Q8⋊D31
►On 248 points
Generators in S248
(1 107 55 78)(2 108 56 79)(3 109 57 80)(4 110 58 81)(5 111 59 82)(6 112 60 83)(7 113 61 84)(8 114 62 85)(9 115 32 86)(10 116 33 87)(11 117 34 88)(12 118 35 89)(13 119 36 90)(14 120 37 91)(15 121 38 92)(16 122 39 93)(17 123 40 63)(18 124 41 64)(19 94 42 65)(20 95 43 66)(21 96 44 67)(22 97 45 68)(23 98 46 69)(24 99 47 70)(25 100 48 71)(26 101 49 72)(27 102 50 73)(28 103 51 74)(29 104 52 75)(30 105 53 76)(31 106 54 77)(125 214 166 239)(126 215 167 240)(127 216 168 241)(128 217 169 242)(129 187 170 243)(130 188 171 244)(131 189 172 245)(132 190 173 246)(133 191 174 247)(134 192 175 248)(135 193 176 218)(136 194 177 219)(137 195 178 220)(138 196 179 221)(139 197 180 222)(140 198 181 223)(141 199 182 224)(142 200 183 225)(143 201 184 226)(144 202 185 227)(145 203 186 228)(146 204 156 229)(147 205 157 230)(148 206 158 231)(149 207 159 232)(150 208 160 233)(151 209 161 234)(152 210 162 235)(153 211 163 236)(154 212 164 237)(155 213 165 238)
(1 177 55 136)(2 178 56 137)(3 179 57 138)(4 180 58 139)(5 181 59 140)(6 182 60 141)(7 183 61 142)(8 184 62 143)(9 185 32 144)(10 186 33 145)(11 156 34 146)(12 157 35 147)(13 158 36 148)(14 159 37 149)(15 160 38 150)(16 161 39 151)(17 162 40 152)(18 163 41 153)(19 164 42 154)(20 165 43 155)(21 166 44 125)(22 167 45 126)(23 168 46 127)(24 169 47 128)(25 170 48 129)(26 171 49 130)(27 172 50 131)(28 173 51 132)(29 174 52 133)(30 175 53 134)(31 176 54 135)(63 235 123 210)(64 236 124 211)(65 237 94 212)(66 238 95 213)(67 239 96 214)(68 240 97 215)(69 241 98 216)(70 242 99 217)(71 243 100 187)(72 244 101 188)(73 245 102 189)(74 246 103 190)(75 247 104 191)(76 248 105 192)(77 218 106 193)(78 219 107 194)(79 220 108 195)(80 221 109 196)(81 222 110 197)(82 223 111 198)(83 224 112 199)(84 225 113 200)(85 226 114 201)(86 227 115 202)(87 228 116 203)(88 229 117 204)(89 230 118 205)(90 231 119 206)(91 232 120 207)(92 233 121 208)(93 234 122 209)
(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)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(47 62)(48 61)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(63 121)(64 120)(65 119)(66 118)(67 117)(68 116)(69 115)(70 114)(71 113)(72 112)(73 111)(74 110)(75 109)(76 108)(77 107)(78 106)(79 105)(80 104)(81 103)(82 102)(83 101)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 124)(92 123)(93 122)(125 204)(126 203)(127 202)(128 201)(129 200)(130 199)(131 198)(132 197)(133 196)(134 195)(135 194)(136 193)(137 192)(138 191)(139 190)(140 189)(141 188)(142 187)(143 217)(144 216)(145 215)(146 214)(147 213)(148 212)(149 211)(150 210)(151 209)(152 208)(153 207)(154 206)(155 205)(156 239)(157 238)(158 237)(159 236)(160 235)(161 234)(162 233)(163 232)(164 231)(165 230)(166 229)(167 228)(168 227)(169 226)(170 225)(171 224)(172 223)(173 222)(174 221)(175 220)(176 219)(177 218)(178 248)(179 247)(180 246)(181 245)(182 244)(183 243)(184 242)(185 241)(186 240)
G:=sub<Sym(248)| (1,107,55,78)(2,108,56,79)(3,109,57,80)(4,110,58,81)(5,111,59,82)(6,112,60,83)(7,113,61,84)(8,114,62,85)(9,115,32,86)(10,116,33,87)(11,117,34,88)(12,118,35,89)(13,119,36,90)(14,120,37,91)(15,121,38,92)(16,122,39,93)(17,123,40,63)(18,124,41,64)(19,94,42,65)(20,95,43,66)(21,96,44,67)(22,97,45,68)(23,98,46,69)(24,99,47,70)(25,100,48,71)(26,101,49,72)(27,102,50,73)(28,103,51,74)(29,104,52,75)(30,105,53,76)(31,106,54,77)(125,214,166,239)(126,215,167,240)(127,216,168,241)(128,217,169,242)(129,187,170,243)(130,188,171,244)(131,189,172,245)(132,190,173,246)(133,191,174,247)(134,192,175,248)(135,193,176,218)(136,194,177,219)(137,195,178,220)(138,196,179,221)(139,197,180,222)(140,198,181,223)(141,199,182,224)(142,200,183,225)(143,201,184,226)(144,202,185,227)(145,203,186,228)(146,204,156,229)(147,205,157,230)(148,206,158,231)(149,207,159,232)(150,208,160,233)(151,209,161,234)(152,210,162,235)(153,211,163,236)(154,212,164,237)(155,213,165,238), (1,177,55,136)(2,178,56,137)(3,179,57,138)(4,180,58,139)(5,181,59,140)(6,182,60,141)(7,183,61,142)(8,184,62,143)(9,185,32,144)(10,186,33,145)(11,156,34,146)(12,157,35,147)(13,158,36,148)(14,159,37,149)(15,160,38,150)(16,161,39,151)(17,162,40,152)(18,163,41,153)(19,164,42,154)(20,165,43,155)(21,166,44,125)(22,167,45,126)(23,168,46,127)(24,169,47,128)(25,170,48,129)(26,171,49,130)(27,172,50,131)(28,173,51,132)(29,174,52,133)(30,175,53,134)(31,176,54,135)(63,235,123,210)(64,236,124,211)(65,237,94,212)(66,238,95,213)(67,239,96,214)(68,240,97,215)(69,241,98,216)(70,242,99,217)(71,243,100,187)(72,244,101,188)(73,245,102,189)(74,246,103,190)(75,247,104,191)(76,248,105,192)(77,218,106,193)(78,219,107,194)(79,220,108,195)(80,221,109,196)(81,222,110,197)(82,223,111,198)(83,224,112,199)(84,225,113,200)(85,226,114,201)(86,227,115,202)(87,228,116,203)(88,229,117,204)(89,230,118,205)(90,231,119,206)(91,232,120,207)(92,233,121,208)(93,234,122,209), (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)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(63,121)(64,120)(65,119)(66,118)(67,117)(68,116)(69,115)(70,114)(71,113)(72,112)(73,111)(74,110)(75,109)(76,108)(77,107)(78,106)(79,105)(80,104)(81,103)(82,102)(83,101)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,124)(92,123)(93,122)(125,204)(126,203)(127,202)(128,201)(129,200)(130,199)(131,198)(132,197)(133,196)(134,195)(135,194)(136,193)(137,192)(138,191)(139,190)(140,189)(141,188)(142,187)(143,217)(144,216)(145,215)(146,214)(147,213)(148,212)(149,211)(150,210)(151,209)(152,208)(153,207)(154,206)(155,205)(156,239)(157,238)(158,237)(159,236)(160,235)(161,234)(162,233)(163,232)(164,231)(165,230)(166,229)(167,228)(168,227)(169,226)(170,225)(171,224)(172,223)(173,222)(174,221)(175,220)(176,219)(177,218)(178,248)(179,247)(180,246)(181,245)(182,244)(183,243)(184,242)(185,241)(186,240)>;
G:=Group( (1,107,55,78)(2,108,56,79)(3,109,57,80)(4,110,58,81)(5,111,59,82)(6,112,60,83)(7,113,61,84)(8,114,62,85)(9,115,32,86)(10,116,33,87)(11,117,34,88)(12,118,35,89)(13,119,36,90)(14,120,37,91)(15,121,38,92)(16,122,39,93)(17,123,40,63)(18,124,41,64)(19,94,42,65)(20,95,43,66)(21,96,44,67)(22,97,45,68)(23,98,46,69)(24,99,47,70)(25,100,48,71)(26,101,49,72)(27,102,50,73)(28,103,51,74)(29,104,52,75)(30,105,53,76)(31,106,54,77)(125,214,166,239)(126,215,167,240)(127,216,168,241)(128,217,169,242)(129,187,170,243)(130,188,171,244)(131,189,172,245)(132,190,173,246)(133,191,174,247)(134,192,175,248)(135,193,176,218)(136,194,177,219)(137,195,178,220)(138,196,179,221)(139,197,180,222)(140,198,181,223)(141,199,182,224)(142,200,183,225)(143,201,184,226)(144,202,185,227)(145,203,186,228)(146,204,156,229)(147,205,157,230)(148,206,158,231)(149,207,159,232)(150,208,160,233)(151,209,161,234)(152,210,162,235)(153,211,163,236)(154,212,164,237)(155,213,165,238), (1,177,55,136)(2,178,56,137)(3,179,57,138)(4,180,58,139)(5,181,59,140)(6,182,60,141)(7,183,61,142)(8,184,62,143)(9,185,32,144)(10,186,33,145)(11,156,34,146)(12,157,35,147)(13,158,36,148)(14,159,37,149)(15,160,38,150)(16,161,39,151)(17,162,40,152)(18,163,41,153)(19,164,42,154)(20,165,43,155)(21,166,44,125)(22,167,45,126)(23,168,46,127)(24,169,47,128)(25,170,48,129)(26,171,49,130)(27,172,50,131)(28,173,51,132)(29,174,52,133)(30,175,53,134)(31,176,54,135)(63,235,123,210)(64,236,124,211)(65,237,94,212)(66,238,95,213)(67,239,96,214)(68,240,97,215)(69,241,98,216)(70,242,99,217)(71,243,100,187)(72,244,101,188)(73,245,102,189)(74,246,103,190)(75,247,104,191)(76,248,105,192)(77,218,106,193)(78,219,107,194)(79,220,108,195)(80,221,109,196)(81,222,110,197)(82,223,111,198)(83,224,112,199)(84,225,113,200)(85,226,114,201)(86,227,115,202)(87,228,116,203)(88,229,117,204)(89,230,118,205)(90,231,119,206)(91,232,120,207)(92,233,121,208)(93,234,122,209), (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)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(63,121)(64,120)(65,119)(66,118)(67,117)(68,116)(69,115)(70,114)(71,113)(72,112)(73,111)(74,110)(75,109)(76,108)(77,107)(78,106)(79,105)(80,104)(81,103)(82,102)(83,101)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,124)(92,123)(93,122)(125,204)(126,203)(127,202)(128,201)(129,200)(130,199)(131,198)(132,197)(133,196)(134,195)(135,194)(136,193)(137,192)(138,191)(139,190)(140,189)(141,188)(142,187)(143,217)(144,216)(145,215)(146,214)(147,213)(148,212)(149,211)(150,210)(151,209)(152,208)(153,207)(154,206)(155,205)(156,239)(157,238)(158,237)(159,236)(160,235)(161,234)(162,233)(163,232)(164,231)(165,230)(166,229)(167,228)(168,227)(169,226)(170,225)(171,224)(172,223)(173,222)(174,221)(175,220)(176,219)(177,218)(178,248)(179,247)(180,246)(181,245)(182,244)(183,243)(184,242)(185,241)(186,240) );
G=PermutationGroup([[(1,107,55,78),(2,108,56,79),(3,109,57,80),(4,110,58,81),(5,111,59,82),(6,112,60,83),(7,113,61,84),(8,114,62,85),(9,115,32,86),(10,116,33,87),(11,117,34,88),(12,118,35,89),(13,119,36,90),(14,120,37,91),(15,121,38,92),(16,122,39,93),(17,123,40,63),(18,124,41,64),(19,94,42,65),(20,95,43,66),(21,96,44,67),(22,97,45,68),(23,98,46,69),(24,99,47,70),(25,100,48,71),(26,101,49,72),(27,102,50,73),(28,103,51,74),(29,104,52,75),(30,105,53,76),(31,106,54,77),(125,214,166,239),(126,215,167,240),(127,216,168,241),(128,217,169,242),(129,187,170,243),(130,188,171,244),(131,189,172,245),(132,190,173,246),(133,191,174,247),(134,192,175,248),(135,193,176,218),(136,194,177,219),(137,195,178,220),(138,196,179,221),(139,197,180,222),(140,198,181,223),(141,199,182,224),(142,200,183,225),(143,201,184,226),(144,202,185,227),(145,203,186,228),(146,204,156,229),(147,205,157,230),(148,206,158,231),(149,207,159,232),(150,208,160,233),(151,209,161,234),(152,210,162,235),(153,211,163,236),(154,212,164,237),(155,213,165,238)], [(1,177,55,136),(2,178,56,137),(3,179,57,138),(4,180,58,139),(5,181,59,140),(6,182,60,141),(7,183,61,142),(8,184,62,143),(9,185,32,144),(10,186,33,145),(11,156,34,146),(12,157,35,147),(13,158,36,148),(14,159,37,149),(15,160,38,150),(16,161,39,151),(17,162,40,152),(18,163,41,153),(19,164,42,154),(20,165,43,155),(21,166,44,125),(22,167,45,126),(23,168,46,127),(24,169,47,128),(25,170,48,129),(26,171,49,130),(27,172,50,131),(28,173,51,132),(29,174,52,133),(30,175,53,134),(31,176,54,135),(63,235,123,210),(64,236,124,211),(65,237,94,212),(66,238,95,213),(67,239,96,214),(68,240,97,215),(69,241,98,216),(70,242,99,217),(71,243,100,187),(72,244,101,188),(73,245,102,189),(74,246,103,190),(75,247,104,191),(76,248,105,192),(77,218,106,193),(78,219,107,194),(79,220,108,195),(80,221,109,196),(81,222,110,197),(82,223,111,198),(83,224,112,199),(84,225,113,200),(85,226,114,201),(86,227,115,202),(87,228,116,203),(88,229,117,204),(89,230,118,205),(90,231,119,206),(91,232,120,207),(92,233,121,208),(93,234,122,209)], [(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),(32,46),(33,45),(34,44),(35,43),(36,42),(37,41),(38,40),(47,62),(48,61),(49,60),(50,59),(51,58),(52,57),(53,56),(54,55),(63,121),(64,120),(65,119),(66,118),(67,117),(68,116),(69,115),(70,114),(71,113),(72,112),(73,111),(74,110),(75,109),(76,108),(77,107),(78,106),(79,105),(80,104),(81,103),(82,102),(83,101),(84,100),(85,99),(86,98),(87,97),(88,96),(89,95),(90,94),(91,124),(92,123),(93,122),(125,204),(126,203),(127,202),(128,201),(129,200),(130,199),(131,198),(132,197),(133,196),(134,195),(135,194),(136,193),(137,192),(138,191),(139,190),(140,189),(141,188),(142,187),(143,217),(144,216),(145,215),(146,214),(147,213),(148,212),(149,211),(150,210),(151,209),(152,208),(153,207),(154,206),(155,205),(156,239),(157,238),(158,237),(159,236),(160,235),(161,234),(162,233),(163,232),(164,231),(165,230),(166,229),(167,228),(168,227),(169,226),(170,225),(171,224),(172,223),(173,222),(174,221),(175,220),(176,219),(177,218),(178,248),(179,247),(180,246),(181,245),(182,244),(183,243),(184,242),(185,241),(186,240)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 31A | ··· | 31O | 62A | ··· | 62O | 124A | ··· | 124AS |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 |
| size | 1 | 1 | 124 | 2 | 4 | 62 | 62 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | D4 | SD16 | D31 | D62 | C31⋊D4 | Q8⋊D31 |
| kernel | Q8⋊D31 | C31⋊C8 | D124 | Q8×C31 | C62 | C31 | Q8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 15 | 15 | 30 | 15 |
Matrix representation of Q8⋊D31 ►in GL4(𝔽1489) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 845 | 278 |
| 0 | 0 | 356 | 644 |
| 1488 | 0 | 0 | 0 |
| 0 | 1488 | 0 | 0 |
| 0 | 0 | 63 | 686 |
| 0 | 0 | 806 | 1426 |
| 1450 | 1 | 0 | 0 |
| 741 | 1050 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 997 | 639 | 0 | 0 |
| 255 | 492 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 326 | 1488 |
G:=sub<GL(4,GF(1489))| [1,0,0,0,0,1,0,0,0,0,845,356,0,0,278,644],[1488,0,0,0,0,1488,0,0,0,0,63,806,0,0,686,1426],[1450,741,0,0,1,1050,0,0,0,0,1,0,0,0,0,1],[997,255,0,0,639,492,0,0,0,0,1,326,0,0,0,1488] >;
Q8⋊D31 in GAP, Magma, Sage, TeX
Q_8\rtimes D_{31} % in TeX
G:=Group("Q8:D31"); // GroupNames label
G:=SmallGroup(496,16);
// by ID
G=gap.SmallGroup(496,16);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-31,61,46,182,97,42,12004]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^31=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations
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