metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.D31, C62.8D4, C4.2D62, C31⋊2SD16, Dic62⋊2C2, C124.2C22, C31⋊C8⋊2C2, (D4×C31).1C2, C2.5(C31⋊D4), SmallGroup(496,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.D31
G = < a,b,c,d | a4=b2=c31=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >
Smallest permutation representation of D4.D31
►On 248 points
Generators in S248
(1 123 61 73)(2 124 62 74)(3 94 32 75)(4 95 33 76)(5 96 34 77)(6 97 35 78)(7 98 36 79)(8 99 37 80)(9 100 38 81)(10 101 39 82)(11 102 40 83)(12 103 41 84)(13 104 42 85)(14 105 43 86)(15 106 44 87)(16 107 45 88)(17 108 46 89)(18 109 47 90)(19 110 48 91)(20 111 49 92)(21 112 50 93)(22 113 51 63)(23 114 52 64)(24 115 53 65)(25 116 54 66)(26 117 55 67)(27 118 56 68)(28 119 57 69)(29 120 58 70)(30 121 59 71)(31 122 60 72)(125 187 156 218)(126 188 157 219)(127 189 158 220)(128 190 159 221)(129 191 160 222)(130 192 161 223)(131 193 162 224)(132 194 163 225)(133 195 164 226)(134 196 165 227)(135 197 166 228)(136 198 167 229)(137 199 168 230)(138 200 169 231)(139 201 170 232)(140 202 171 233)(141 203 172 234)(142 204 173 235)(143 205 174 236)(144 206 175 237)(145 207 176 238)(146 208 177 239)(147 209 178 240)(148 210 179 241)(149 211 180 242)(150 212 181 243)(151 213 182 244)(152 214 183 245)(153 215 184 246)(154 216 185 247)(155 217 186 248)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 63)(23 64)(24 65)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 94)(33 95)(34 96)(35 97)(36 98)(37 99)(38 100)(39 101)(40 102)(41 103)(42 104)(43 105)(44 106)(45 107)(46 108)(47 109)(48 110)(49 111)(50 112)(51 113)(52 114)(53 115)(54 116)(55 117)(56 118)(57 119)(58 120)(59 121)(60 122)(61 123)(62 124)(125 156)(126 157)(127 158)(128 159)(129 160)(130 161)(131 162)(132 163)(133 164)(134 165)(135 166)(136 167)(137 168)(138 169)(139 170)(140 171)(141 172)(142 173)(143 174)(144 175)(145 176)(146 177)(147 178)(148 179)(149 180)(150 181)(151 182)(152 183)(153 184)(154 185)(155 186)
(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)
(1 186 61 155)(2 185 62 154)(3 184 32 153)(4 183 33 152)(5 182 34 151)(6 181 35 150)(7 180 36 149)(8 179 37 148)(9 178 38 147)(10 177 39 146)(11 176 40 145)(12 175 41 144)(13 174 42 143)(14 173 43 142)(15 172 44 141)(16 171 45 140)(17 170 46 139)(18 169 47 138)(19 168 48 137)(20 167 49 136)(21 166 50 135)(22 165 51 134)(23 164 52 133)(24 163 53 132)(25 162 54 131)(26 161 55 130)(27 160 56 129)(28 159 57 128)(29 158 58 127)(30 157 59 126)(31 156 60 125)(63 227 113 196)(64 226 114 195)(65 225 115 194)(66 224 116 193)(67 223 117 192)(68 222 118 191)(69 221 119 190)(70 220 120 189)(71 219 121 188)(72 218 122 187)(73 248 123 217)(74 247 124 216)(75 246 94 215)(76 245 95 214)(77 244 96 213)(78 243 97 212)(79 242 98 211)(80 241 99 210)(81 240 100 209)(82 239 101 208)(83 238 102 207)(84 237 103 206)(85 236 104 205)(86 235 105 204)(87 234 106 203)(88 233 107 202)(89 232 108 201)(90 231 109 200)(91 230 110 199)(92 229 111 198)(93 228 112 197)
G:=sub<Sym(248)| (1,123,61,73)(2,124,62,74)(3,94,32,75)(4,95,33,76)(5,96,34,77)(6,97,35,78)(7,98,36,79)(8,99,37,80)(9,100,38,81)(10,101,39,82)(11,102,40,83)(12,103,41,84)(13,104,42,85)(14,105,43,86)(15,106,44,87)(16,107,45,88)(17,108,46,89)(18,109,47,90)(19,110,48,91)(20,111,49,92)(21,112,50,93)(22,113,51,63)(23,114,52,64)(24,115,53,65)(25,116,54,66)(26,117,55,67)(27,118,56,68)(28,119,57,69)(29,120,58,70)(30,121,59,71)(31,122,60,72)(125,187,156,218)(126,188,157,219)(127,189,158,220)(128,190,159,221)(129,191,160,222)(130,192,161,223)(131,193,162,224)(132,194,163,225)(133,195,164,226)(134,196,165,227)(135,197,166,228)(136,198,167,229)(137,199,168,230)(138,200,169,231)(139,201,170,232)(140,202,171,233)(141,203,172,234)(142,204,173,235)(143,205,174,236)(144,206,175,237)(145,207,176,238)(146,208,177,239)(147,209,178,240)(148,210,179,241)(149,211,180,242)(150,212,181,243)(151,213,182,244)(152,214,183,245)(153,215,184,246)(154,216,185,247)(155,217,186,248), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,63)(23,64)(24,65)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,94)(33,95)(34,96)(35,97)(36,98)(37,99)(38,100)(39,101)(40,102)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,109)(48,110)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,121)(60,122)(61,123)(62,124)(125,156)(126,157)(127,158)(128,159)(129,160)(130,161)(131,162)(132,163)(133,164)(134,165)(135,166)(136,167)(137,168)(138,169)(139,170)(140,171)(141,172)(142,173)(143,174)(144,175)(145,176)(146,177)(147,178)(148,179)(149,180)(150,181)(151,182)(152,183)(153,184)(154,185)(155,186), (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), (1,186,61,155)(2,185,62,154)(3,184,32,153)(4,183,33,152)(5,182,34,151)(6,181,35,150)(7,180,36,149)(8,179,37,148)(9,178,38,147)(10,177,39,146)(11,176,40,145)(12,175,41,144)(13,174,42,143)(14,173,43,142)(15,172,44,141)(16,171,45,140)(17,170,46,139)(18,169,47,138)(19,168,48,137)(20,167,49,136)(21,166,50,135)(22,165,51,134)(23,164,52,133)(24,163,53,132)(25,162,54,131)(26,161,55,130)(27,160,56,129)(28,159,57,128)(29,158,58,127)(30,157,59,126)(31,156,60,125)(63,227,113,196)(64,226,114,195)(65,225,115,194)(66,224,116,193)(67,223,117,192)(68,222,118,191)(69,221,119,190)(70,220,120,189)(71,219,121,188)(72,218,122,187)(73,248,123,217)(74,247,124,216)(75,246,94,215)(76,245,95,214)(77,244,96,213)(78,243,97,212)(79,242,98,211)(80,241,99,210)(81,240,100,209)(82,239,101,208)(83,238,102,207)(84,237,103,206)(85,236,104,205)(86,235,105,204)(87,234,106,203)(88,233,107,202)(89,232,108,201)(90,231,109,200)(91,230,110,199)(92,229,111,198)(93,228,112,197)>;
G:=Group( (1,123,61,73)(2,124,62,74)(3,94,32,75)(4,95,33,76)(5,96,34,77)(6,97,35,78)(7,98,36,79)(8,99,37,80)(9,100,38,81)(10,101,39,82)(11,102,40,83)(12,103,41,84)(13,104,42,85)(14,105,43,86)(15,106,44,87)(16,107,45,88)(17,108,46,89)(18,109,47,90)(19,110,48,91)(20,111,49,92)(21,112,50,93)(22,113,51,63)(23,114,52,64)(24,115,53,65)(25,116,54,66)(26,117,55,67)(27,118,56,68)(28,119,57,69)(29,120,58,70)(30,121,59,71)(31,122,60,72)(125,187,156,218)(126,188,157,219)(127,189,158,220)(128,190,159,221)(129,191,160,222)(130,192,161,223)(131,193,162,224)(132,194,163,225)(133,195,164,226)(134,196,165,227)(135,197,166,228)(136,198,167,229)(137,199,168,230)(138,200,169,231)(139,201,170,232)(140,202,171,233)(141,203,172,234)(142,204,173,235)(143,205,174,236)(144,206,175,237)(145,207,176,238)(146,208,177,239)(147,209,178,240)(148,210,179,241)(149,211,180,242)(150,212,181,243)(151,213,182,244)(152,214,183,245)(153,215,184,246)(154,216,185,247)(155,217,186,248), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,63)(23,64)(24,65)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,94)(33,95)(34,96)(35,97)(36,98)(37,99)(38,100)(39,101)(40,102)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,109)(48,110)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,121)(60,122)(61,123)(62,124)(125,156)(126,157)(127,158)(128,159)(129,160)(130,161)(131,162)(132,163)(133,164)(134,165)(135,166)(136,167)(137,168)(138,169)(139,170)(140,171)(141,172)(142,173)(143,174)(144,175)(145,176)(146,177)(147,178)(148,179)(149,180)(150,181)(151,182)(152,183)(153,184)(154,185)(155,186), (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), (1,186,61,155)(2,185,62,154)(3,184,32,153)(4,183,33,152)(5,182,34,151)(6,181,35,150)(7,180,36,149)(8,179,37,148)(9,178,38,147)(10,177,39,146)(11,176,40,145)(12,175,41,144)(13,174,42,143)(14,173,43,142)(15,172,44,141)(16,171,45,140)(17,170,46,139)(18,169,47,138)(19,168,48,137)(20,167,49,136)(21,166,50,135)(22,165,51,134)(23,164,52,133)(24,163,53,132)(25,162,54,131)(26,161,55,130)(27,160,56,129)(28,159,57,128)(29,158,58,127)(30,157,59,126)(31,156,60,125)(63,227,113,196)(64,226,114,195)(65,225,115,194)(66,224,116,193)(67,223,117,192)(68,222,118,191)(69,221,119,190)(70,220,120,189)(71,219,121,188)(72,218,122,187)(73,248,123,217)(74,247,124,216)(75,246,94,215)(76,245,95,214)(77,244,96,213)(78,243,97,212)(79,242,98,211)(80,241,99,210)(81,240,100,209)(82,239,101,208)(83,238,102,207)(84,237,103,206)(85,236,104,205)(86,235,105,204)(87,234,106,203)(88,233,107,202)(89,232,108,201)(90,231,109,200)(91,230,110,199)(92,229,111,198)(93,228,112,197) );
G=PermutationGroup([[(1,123,61,73),(2,124,62,74),(3,94,32,75),(4,95,33,76),(5,96,34,77),(6,97,35,78),(7,98,36,79),(8,99,37,80),(9,100,38,81),(10,101,39,82),(11,102,40,83),(12,103,41,84),(13,104,42,85),(14,105,43,86),(15,106,44,87),(16,107,45,88),(17,108,46,89),(18,109,47,90),(19,110,48,91),(20,111,49,92),(21,112,50,93),(22,113,51,63),(23,114,52,64),(24,115,53,65),(25,116,54,66),(26,117,55,67),(27,118,56,68),(28,119,57,69),(29,120,58,70),(30,121,59,71),(31,122,60,72),(125,187,156,218),(126,188,157,219),(127,189,158,220),(128,190,159,221),(129,191,160,222),(130,192,161,223),(131,193,162,224),(132,194,163,225),(133,195,164,226),(134,196,165,227),(135,197,166,228),(136,198,167,229),(137,199,168,230),(138,200,169,231),(139,201,170,232),(140,202,171,233),(141,203,172,234),(142,204,173,235),(143,205,174,236),(144,206,175,237),(145,207,176,238),(146,208,177,239),(147,209,178,240),(148,210,179,241),(149,211,180,242),(150,212,181,243),(151,213,182,244),(152,214,183,245),(153,215,184,246),(154,216,185,247),(155,217,186,248)], [(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,79),(8,80),(9,81),(10,82),(11,83),(12,84),(13,85),(14,86),(15,87),(16,88),(17,89),(18,90),(19,91),(20,92),(21,93),(22,63),(23,64),(24,65),(25,66),(26,67),(27,68),(28,69),(29,70),(30,71),(31,72),(32,94),(33,95),(34,96),(35,97),(36,98),(37,99),(38,100),(39,101),(40,102),(41,103),(42,104),(43,105),(44,106),(45,107),(46,108),(47,109),(48,110),(49,111),(50,112),(51,113),(52,114),(53,115),(54,116),(55,117),(56,118),(57,119),(58,120),(59,121),(60,122),(61,123),(62,124),(125,156),(126,157),(127,158),(128,159),(129,160),(130,161),(131,162),(132,163),(133,164),(134,165),(135,166),(136,167),(137,168),(138,169),(139,170),(140,171),(141,172),(142,173),(143,174),(144,175),(145,176),(146,177),(147,178),(148,179),(149,180),(150,181),(151,182),(152,183),(153,184),(154,185),(155,186)], [(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)], [(1,186,61,155),(2,185,62,154),(3,184,32,153),(4,183,33,152),(5,182,34,151),(6,181,35,150),(7,180,36,149),(8,179,37,148),(9,178,38,147),(10,177,39,146),(11,176,40,145),(12,175,41,144),(13,174,42,143),(14,173,43,142),(15,172,44,141),(16,171,45,140),(17,170,46,139),(18,169,47,138),(19,168,48,137),(20,167,49,136),(21,166,50,135),(22,165,51,134),(23,164,52,133),(24,163,53,132),(25,162,54,131),(26,161,55,130),(27,160,56,129),(28,159,57,128),(29,158,58,127),(30,157,59,126),(31,156,60,125),(63,227,113,196),(64,226,114,195),(65,225,115,194),(66,224,116,193),(67,223,117,192),(68,222,118,191),(69,221,119,190),(70,220,120,189),(71,219,121,188),(72,218,122,187),(73,248,123,217),(74,247,124,216),(75,246,94,215),(76,245,95,214),(77,244,96,213),(78,243,97,212),(79,242,98,211),(80,241,99,210),(81,240,100,209),(82,239,101,208),(83,238,102,207),(84,237,103,206),(85,236,104,205),(86,235,105,204),(87,234,106,203),(88,233,107,202),(89,232,108,201),(90,231,109,200),(91,230,110,199),(92,229,111,198),(93,228,112,197)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 31A | ··· | 31O | 62A | ··· | 62O | 62P | ··· | 62AS | 124A | ··· | 124O |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 31 | ··· | 31 | 62 | ··· | 62 | 62 | ··· | 62 | 124 | ··· | 124 |
| size | 1 | 1 | 4 | 2 | 124 | 62 | 62 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 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 | D4.D31 |
| kernel | D4.D31 | C31⋊C8 | Dic62 | D4×C31 | C62 | C31 | D4 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 15 | 15 | 30 | 15 |
Matrix representation of D4.D31 ►in GL4(𝔽1489) generated by
| 1488 | 0 | 0 | 0 |
| 0 | 1488 | 0 | 0 |
| 0 | 0 | 1 | 296 |
| 0 | 0 | 332 | 1488 |
| 1488 | 868 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 296 |
| 0 | 0 | 0 | 1488 |
| 1401 | 1402 | 0 | 0 |
| 0 | 423 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1401 | 1402 | 0 | 0 |
| 89 | 88 | 0 | 0 |
| 0 | 0 | 0 | 73 |
| 0 | 0 | 1387 | 0 |
G:=sub<GL(4,GF(1489))| [1488,0,0,0,0,1488,0,0,0,0,1,332,0,0,296,1488],[1488,0,0,0,868,1,0,0,0,0,1,0,0,0,296,1488],[1401,0,0,0,1402,423,0,0,0,0,1,0,0,0,0,1],[1401,89,0,0,1402,88,0,0,0,0,0,1387,0,0,73,0] >;
D4.D31 in GAP, Magma, Sage, TeX
D_4.D_{31} % in TeX
G:=Group("D4.D31"); // GroupNames label
G:=SmallGroup(496,15);
// by ID
G=gap.SmallGroup(496,15);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-31,40,61,182,97,42,12004]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^31=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations
Export