metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊2D31, C4.7D62, D124⋊4C2, C62.8C23, C124.7C22, D62.3C22, Dic31.5C22, (C4×D31)⋊3C2, C31⋊3(C4○D4), (Q8×C31)⋊3C2, C2.9(C22×D31), SmallGroup(496,34)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊2D31
G = < a,b,c,d | a4=c31=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of Q8⋊2D31
►On 248 points
Generators in S248
(1 101 38 71)(2 102 39 72)(3 103 40 73)(4 104 41 74)(5 105 42 75)(6 106 43 76)(7 107 44 77)(8 108 45 78)(9 109 46 79)(10 110 47 80)(11 111 48 81)(12 112 49 82)(13 113 50 83)(14 114 51 84)(15 115 52 85)(16 116 53 86)(17 117 54 87)(18 118 55 88)(19 119 56 89)(20 120 57 90)(21 121 58 91)(22 122 59 92)(23 123 60 93)(24 124 61 63)(25 94 62 64)(26 95 32 65)(27 96 33 66)(28 97 34 67)(29 98 35 68)(30 99 36 69)(31 100 37 70)(125 190 176 241)(126 191 177 242)(127 192 178 243)(128 193 179 244)(129 194 180 245)(130 195 181 246)(131 196 182 247)(132 197 183 248)(133 198 184 218)(134 199 185 219)(135 200 186 220)(136 201 156 221)(137 202 157 222)(138 203 158 223)(139 204 159 224)(140 205 160 225)(141 206 161 226)(142 207 162 227)(143 208 163 228)(144 209 164 229)(145 210 165 230)(146 211 166 231)(147 212 167 232)(148 213 168 233)(149 214 169 234)(150 215 170 235)(151 216 171 236)(152 217 172 237)(153 187 173 238)(154 188 174 239)(155 189 175 240)
(1 186 38 135)(2 156 39 136)(3 157 40 137)(4 158 41 138)(5 159 42 139)(6 160 43 140)(7 161 44 141)(8 162 45 142)(9 163 46 143)(10 164 47 144)(11 165 48 145)(12 166 49 146)(13 167 50 147)(14 168 51 148)(15 169 52 149)(16 170 53 150)(17 171 54 151)(18 172 55 152)(19 173 56 153)(20 174 57 154)(21 175 58 155)(22 176 59 125)(23 177 60 126)(24 178 61 127)(25 179 62 128)(26 180 32 129)(27 181 33 130)(28 182 34 131)(29 183 35 132)(30 184 36 133)(31 185 37 134)(63 243 124 192)(64 244 94 193)(65 245 95 194)(66 246 96 195)(67 247 97 196)(68 248 98 197)(69 218 99 198)(70 219 100 199)(71 220 101 200)(72 221 102 201)(73 222 103 202)(74 223 104 203)(75 224 105 204)(76 225 106 205)(77 226 107 206)(78 227 108 207)(79 228 109 208)(80 229 110 209)(81 230 111 210)(82 231 112 211)(83 232 113 212)(84 233 114 213)(85 234 115 214)(86 235 116 215)(87 236 117 216)(88 237 118 217)(89 238 119 187)(90 239 120 188)(91 240 121 189)(92 241 122 190)(93 242 123 191)
(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 70)(2 69)(3 68)(4 67)(5 66)(6 65)(7 64)(8 63)(9 93)(10 92)(11 91)(12 90)(13 89)(14 88)(15 87)(16 86)(17 85)(18 84)(19 83)(20 82)(21 81)(22 80)(23 79)(24 78)(25 77)(26 76)(27 75)(28 74)(29 73)(30 72)(31 71)(32 106)(33 105)(34 104)(35 103)(36 102)(37 101)(38 100)(39 99)(40 98)(41 97)(42 96)(43 95)(44 94)(45 124)(46 123)(47 122)(48 121)(49 120)(50 119)(51 118)(52 117)(53 116)(54 115)(55 114)(56 113)(57 112)(58 111)(59 110)(60 109)(61 108)(62 107)(125 209)(126 208)(127 207)(128 206)(129 205)(130 204)(131 203)(132 202)(133 201)(134 200)(135 199)(136 198)(137 197)(138 196)(139 195)(140 194)(141 193)(142 192)(143 191)(144 190)(145 189)(146 188)(147 187)(148 217)(149 216)(150 215)(151 214)(152 213)(153 212)(154 211)(155 210)(156 218)(157 248)(158 247)(159 246)(160 245)(161 244)(162 243)(163 242)(164 241)(165 240)(166 239)(167 238)(168 237)(169 236)(170 235)(171 234)(172 233)(173 232)(174 231)(175 230)(176 229)(177 228)(178 227)(179 226)(180 225)(181 224)(182 223)(183 222)(184 221)(185 220)(186 219)
G:=sub<Sym(248)| (1,101,38,71)(2,102,39,72)(3,103,40,73)(4,104,41,74)(5,105,42,75)(6,106,43,76)(7,107,44,77)(8,108,45,78)(9,109,46,79)(10,110,47,80)(11,111,48,81)(12,112,49,82)(13,113,50,83)(14,114,51,84)(15,115,52,85)(16,116,53,86)(17,117,54,87)(18,118,55,88)(19,119,56,89)(20,120,57,90)(21,121,58,91)(22,122,59,92)(23,123,60,93)(24,124,61,63)(25,94,62,64)(26,95,32,65)(27,96,33,66)(28,97,34,67)(29,98,35,68)(30,99,36,69)(31,100,37,70)(125,190,176,241)(126,191,177,242)(127,192,178,243)(128,193,179,244)(129,194,180,245)(130,195,181,246)(131,196,182,247)(132,197,183,248)(133,198,184,218)(134,199,185,219)(135,200,186,220)(136,201,156,221)(137,202,157,222)(138,203,158,223)(139,204,159,224)(140,205,160,225)(141,206,161,226)(142,207,162,227)(143,208,163,228)(144,209,164,229)(145,210,165,230)(146,211,166,231)(147,212,167,232)(148,213,168,233)(149,214,169,234)(150,215,170,235)(151,216,171,236)(152,217,172,237)(153,187,173,238)(154,188,174,239)(155,189,175,240), (1,186,38,135)(2,156,39,136)(3,157,40,137)(4,158,41,138)(5,159,42,139)(6,160,43,140)(7,161,44,141)(8,162,45,142)(9,163,46,143)(10,164,47,144)(11,165,48,145)(12,166,49,146)(13,167,50,147)(14,168,51,148)(15,169,52,149)(16,170,53,150)(17,171,54,151)(18,172,55,152)(19,173,56,153)(20,174,57,154)(21,175,58,155)(22,176,59,125)(23,177,60,126)(24,178,61,127)(25,179,62,128)(26,180,32,129)(27,181,33,130)(28,182,34,131)(29,183,35,132)(30,184,36,133)(31,185,37,134)(63,243,124,192)(64,244,94,193)(65,245,95,194)(66,246,96,195)(67,247,97,196)(68,248,98,197)(69,218,99,198)(70,219,100,199)(71,220,101,200)(72,221,102,201)(73,222,103,202)(74,223,104,203)(75,224,105,204)(76,225,106,205)(77,226,107,206)(78,227,108,207)(79,228,109,208)(80,229,110,209)(81,230,111,210)(82,231,112,211)(83,232,113,212)(84,233,114,213)(85,234,115,214)(86,235,116,215)(87,236,117,216)(88,237,118,217)(89,238,119,187)(90,239,120,188)(91,240,121,189)(92,241,122,190)(93,242,123,191), (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), (1,70)(2,69)(3,68)(4,67)(5,66)(6,65)(7,64)(8,63)(9,93)(10,92)(11,91)(12,90)(13,89)(14,88)(15,87)(16,86)(17,85)(18,84)(19,83)(20,82)(21,81)(22,80)(23,79)(24,78)(25,77)(26,76)(27,75)(28,74)(29,73)(30,72)(31,71)(32,106)(33,105)(34,104)(35,103)(36,102)(37,101)(38,100)(39,99)(40,98)(41,97)(42,96)(43,95)(44,94)(45,124)(46,123)(47,122)(48,121)(49,120)(50,119)(51,118)(52,117)(53,116)(54,115)(55,114)(56,113)(57,112)(58,111)(59,110)(60,109)(61,108)(62,107)(125,209)(126,208)(127,207)(128,206)(129,205)(130,204)(131,203)(132,202)(133,201)(134,200)(135,199)(136,198)(137,197)(138,196)(139,195)(140,194)(141,193)(142,192)(143,191)(144,190)(145,189)(146,188)(147,187)(148,217)(149,216)(150,215)(151,214)(152,213)(153,212)(154,211)(155,210)(156,218)(157,248)(158,247)(159,246)(160,245)(161,244)(162,243)(163,242)(164,241)(165,240)(166,239)(167,238)(168,237)(169,236)(170,235)(171,234)(172,233)(173,232)(174,231)(175,230)(176,229)(177,228)(178,227)(179,226)(180,225)(181,224)(182,223)(183,222)(184,221)(185,220)(186,219)>;
G:=Group( (1,101,38,71)(2,102,39,72)(3,103,40,73)(4,104,41,74)(5,105,42,75)(6,106,43,76)(7,107,44,77)(8,108,45,78)(9,109,46,79)(10,110,47,80)(11,111,48,81)(12,112,49,82)(13,113,50,83)(14,114,51,84)(15,115,52,85)(16,116,53,86)(17,117,54,87)(18,118,55,88)(19,119,56,89)(20,120,57,90)(21,121,58,91)(22,122,59,92)(23,123,60,93)(24,124,61,63)(25,94,62,64)(26,95,32,65)(27,96,33,66)(28,97,34,67)(29,98,35,68)(30,99,36,69)(31,100,37,70)(125,190,176,241)(126,191,177,242)(127,192,178,243)(128,193,179,244)(129,194,180,245)(130,195,181,246)(131,196,182,247)(132,197,183,248)(133,198,184,218)(134,199,185,219)(135,200,186,220)(136,201,156,221)(137,202,157,222)(138,203,158,223)(139,204,159,224)(140,205,160,225)(141,206,161,226)(142,207,162,227)(143,208,163,228)(144,209,164,229)(145,210,165,230)(146,211,166,231)(147,212,167,232)(148,213,168,233)(149,214,169,234)(150,215,170,235)(151,216,171,236)(152,217,172,237)(153,187,173,238)(154,188,174,239)(155,189,175,240), (1,186,38,135)(2,156,39,136)(3,157,40,137)(4,158,41,138)(5,159,42,139)(6,160,43,140)(7,161,44,141)(8,162,45,142)(9,163,46,143)(10,164,47,144)(11,165,48,145)(12,166,49,146)(13,167,50,147)(14,168,51,148)(15,169,52,149)(16,170,53,150)(17,171,54,151)(18,172,55,152)(19,173,56,153)(20,174,57,154)(21,175,58,155)(22,176,59,125)(23,177,60,126)(24,178,61,127)(25,179,62,128)(26,180,32,129)(27,181,33,130)(28,182,34,131)(29,183,35,132)(30,184,36,133)(31,185,37,134)(63,243,124,192)(64,244,94,193)(65,245,95,194)(66,246,96,195)(67,247,97,196)(68,248,98,197)(69,218,99,198)(70,219,100,199)(71,220,101,200)(72,221,102,201)(73,222,103,202)(74,223,104,203)(75,224,105,204)(76,225,106,205)(77,226,107,206)(78,227,108,207)(79,228,109,208)(80,229,110,209)(81,230,111,210)(82,231,112,211)(83,232,113,212)(84,233,114,213)(85,234,115,214)(86,235,116,215)(87,236,117,216)(88,237,118,217)(89,238,119,187)(90,239,120,188)(91,240,121,189)(92,241,122,190)(93,242,123,191), (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), (1,70)(2,69)(3,68)(4,67)(5,66)(6,65)(7,64)(8,63)(9,93)(10,92)(11,91)(12,90)(13,89)(14,88)(15,87)(16,86)(17,85)(18,84)(19,83)(20,82)(21,81)(22,80)(23,79)(24,78)(25,77)(26,76)(27,75)(28,74)(29,73)(30,72)(31,71)(32,106)(33,105)(34,104)(35,103)(36,102)(37,101)(38,100)(39,99)(40,98)(41,97)(42,96)(43,95)(44,94)(45,124)(46,123)(47,122)(48,121)(49,120)(50,119)(51,118)(52,117)(53,116)(54,115)(55,114)(56,113)(57,112)(58,111)(59,110)(60,109)(61,108)(62,107)(125,209)(126,208)(127,207)(128,206)(129,205)(130,204)(131,203)(132,202)(133,201)(134,200)(135,199)(136,198)(137,197)(138,196)(139,195)(140,194)(141,193)(142,192)(143,191)(144,190)(145,189)(146,188)(147,187)(148,217)(149,216)(150,215)(151,214)(152,213)(153,212)(154,211)(155,210)(156,218)(157,248)(158,247)(159,246)(160,245)(161,244)(162,243)(163,242)(164,241)(165,240)(166,239)(167,238)(168,237)(169,236)(170,235)(171,234)(172,233)(173,232)(174,231)(175,230)(176,229)(177,228)(178,227)(179,226)(180,225)(181,224)(182,223)(183,222)(184,221)(185,220)(186,219) );
G=PermutationGroup([[(1,101,38,71),(2,102,39,72),(3,103,40,73),(4,104,41,74),(5,105,42,75),(6,106,43,76),(7,107,44,77),(8,108,45,78),(9,109,46,79),(10,110,47,80),(11,111,48,81),(12,112,49,82),(13,113,50,83),(14,114,51,84),(15,115,52,85),(16,116,53,86),(17,117,54,87),(18,118,55,88),(19,119,56,89),(20,120,57,90),(21,121,58,91),(22,122,59,92),(23,123,60,93),(24,124,61,63),(25,94,62,64),(26,95,32,65),(27,96,33,66),(28,97,34,67),(29,98,35,68),(30,99,36,69),(31,100,37,70),(125,190,176,241),(126,191,177,242),(127,192,178,243),(128,193,179,244),(129,194,180,245),(130,195,181,246),(131,196,182,247),(132,197,183,248),(133,198,184,218),(134,199,185,219),(135,200,186,220),(136,201,156,221),(137,202,157,222),(138,203,158,223),(139,204,159,224),(140,205,160,225),(141,206,161,226),(142,207,162,227),(143,208,163,228),(144,209,164,229),(145,210,165,230),(146,211,166,231),(147,212,167,232),(148,213,168,233),(149,214,169,234),(150,215,170,235),(151,216,171,236),(152,217,172,237),(153,187,173,238),(154,188,174,239),(155,189,175,240)], [(1,186,38,135),(2,156,39,136),(3,157,40,137),(4,158,41,138),(5,159,42,139),(6,160,43,140),(7,161,44,141),(8,162,45,142),(9,163,46,143),(10,164,47,144),(11,165,48,145),(12,166,49,146),(13,167,50,147),(14,168,51,148),(15,169,52,149),(16,170,53,150),(17,171,54,151),(18,172,55,152),(19,173,56,153),(20,174,57,154),(21,175,58,155),(22,176,59,125),(23,177,60,126),(24,178,61,127),(25,179,62,128),(26,180,32,129),(27,181,33,130),(28,182,34,131),(29,183,35,132),(30,184,36,133),(31,185,37,134),(63,243,124,192),(64,244,94,193),(65,245,95,194),(66,246,96,195),(67,247,97,196),(68,248,98,197),(69,218,99,198),(70,219,100,199),(71,220,101,200),(72,221,102,201),(73,222,103,202),(74,223,104,203),(75,224,105,204),(76,225,106,205),(77,226,107,206),(78,227,108,207),(79,228,109,208),(80,229,110,209),(81,230,111,210),(82,231,112,211),(83,232,113,212),(84,233,114,213),(85,234,115,214),(86,235,116,215),(87,236,117,216),(88,237,118,217),(89,238,119,187),(90,239,120,188),(91,240,121,189),(92,241,122,190),(93,242,123,191)], [(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)], [(1,70),(2,69),(3,68),(4,67),(5,66),(6,65),(7,64),(8,63),(9,93),(10,92),(11,91),(12,90),(13,89),(14,88),(15,87),(16,86),(17,85),(18,84),(19,83),(20,82),(21,81),(22,80),(23,79),(24,78),(25,77),(26,76),(27,75),(28,74),(29,73),(30,72),(31,71),(32,106),(33,105),(34,104),(35,103),(36,102),(37,101),(38,100),(39,99),(40,98),(41,97),(42,96),(43,95),(44,94),(45,124),(46,123),(47,122),(48,121),(49,120),(50,119),(51,118),(52,117),(53,116),(54,115),(55,114),(56,113),(57,112),(58,111),(59,110),(60,109),(61,108),(62,107),(125,209),(126,208),(127,207),(128,206),(129,205),(130,204),(131,203),(132,202),(133,201),(134,200),(135,199),(136,198),(137,197),(138,196),(139,195),(140,194),(141,193),(142,192),(143,191),(144,190),(145,189),(146,188),(147,187),(148,217),(149,216),(150,215),(151,214),(152,213),(153,212),(154,211),(155,210),(156,218),(157,248),(158,247),(159,246),(160,245),(161,244),(162,243),(163,242),(164,241),(165,240),(166,239),(167,238),(168,237),(169,236),(170,235),(171,234),(172,233),(173,232),(174,231),(175,230),(176,229),(177,228),(178,227),(179,226),(180,225),(181,224),(182,223),(183,222),(184,221),(185,220),(186,219)]])
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 31A | ··· | 31O | 62A | ··· | 62O | 124A | ··· | 124AS |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 |
| size | 1 | 1 | 62 | 62 | 62 | 2 | 2 | 2 | 31 | 31 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C4○D4 | D31 | D62 | Q8⋊2D31 |
| kernel | Q8⋊2D31 | C4×D31 | D124 | Q8×C31 | C31 | Q8 | C4 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 15 | 45 | 15 |
Matrix representation of Q8⋊2D31 ►in GL4(𝔽373) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 372 | 0 |
| 372 | 0 | 0 | 0 |
| 0 | 372 | 0 | 0 |
| 0 | 0 | 0 | 269 |
| 0 | 0 | 269 | 0 |
| 88 | 1 | 0 | 0 |
| 171 | 358 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 358 | 372 | 0 | 0 |
| 224 | 15 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(373))| [1,0,0,0,0,1,0,0,0,0,0,372,0,0,1,0],[372,0,0,0,0,372,0,0,0,0,0,269,0,0,269,0],[88,171,0,0,1,358,0,0,0,0,1,0,0,0,0,1],[358,224,0,0,372,15,0,0,0,0,0,1,0,0,1,0] >;
Q8⋊2D31 in GAP, Magma, Sage, TeX
Q_8\rtimes_2D_{31} % in TeX
G:=Group("Q8:2D31"); // GroupNames label
G:=SmallGroup(496,34);
// by ID
G=gap.SmallGroup(496,34);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-31,46,182,97,42,12004]);
// 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,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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