direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: SD16×C31, Q8⋊C62, C8⋊2C62, D4.C62, C248⋊6C2, C62.15D4, C124.18C22, C4.2(C2×C62), (Q8×C31)⋊4C2, C2.4(D4×C31), (D4×C31).2C2, SmallGroup(496,25)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16×C31
G = < a,b,c | a31=b8=c2=1, ab=ba, ac=ca, cbc=b3 >
Smallest permutation representation of SD16×C31
►On 248 points
Generators in S248
(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)
(1 120 33 201 76 177 127 237)(2 121 34 202 77 178 128 238)(3 122 35 203 78 179 129 239)(4 123 36 204 79 180 130 240)(5 124 37 205 80 181 131 241)(6 94 38 206 81 182 132 242)(7 95 39 207 82 183 133 243)(8 96 40 208 83 184 134 244)(9 97 41 209 84 185 135 245)(10 98 42 210 85 186 136 246)(11 99 43 211 86 156 137 247)(12 100 44 212 87 157 138 248)(13 101 45 213 88 158 139 218)(14 102 46 214 89 159 140 219)(15 103 47 215 90 160 141 220)(16 104 48 216 91 161 142 221)(17 105 49 217 92 162 143 222)(18 106 50 187 93 163 144 223)(19 107 51 188 63 164 145 224)(20 108 52 189 64 165 146 225)(21 109 53 190 65 166 147 226)(22 110 54 191 66 167 148 227)(23 111 55 192 67 168 149 228)(24 112 56 193 68 169 150 229)(25 113 57 194 69 170 151 230)(26 114 58 195 70 171 152 231)(27 115 59 196 71 172 153 232)(28 116 60 197 72 173 154 233)(29 117 61 198 73 174 155 234)(30 118 62 199 74 175 125 235)(31 119 32 200 75 176 126 236)
(32 126)(33 127)(34 128)(35 129)(36 130)(37 131)(38 132)(39 133)(40 134)(41 135)(42 136)(43 137)(44 138)(45 139)(46 140)(47 141)(48 142)(49 143)(50 144)(51 145)(52 146)(53 147)(54 148)(55 149)(56 150)(57 151)(58 152)(59 153)(60 154)(61 155)(62 125)(94 206)(95 207)(96 208)(97 209)(98 210)(99 211)(100 212)(101 213)(102 214)(103 215)(104 216)(105 217)(106 187)(107 188)(108 189)(109 190)(110 191)(111 192)(112 193)(113 194)(114 195)(115 196)(116 197)(117 198)(118 199)(119 200)(120 201)(121 202)(122 203)(123 204)(124 205)(156 247)(157 248)(158 218)(159 219)(160 220)(161 221)(162 222)(163 223)(164 224)(165 225)(166 226)(167 227)(168 228)(169 229)(170 230)(171 231)(172 232)(173 233)(174 234)(175 235)(176 236)(177 237)(178 238)(179 239)(180 240)(181 241)(182 242)(183 243)(184 244)(185 245)(186 246)
G:=sub<Sym(248)| (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), (1,120,33,201,76,177,127,237)(2,121,34,202,77,178,128,238)(3,122,35,203,78,179,129,239)(4,123,36,204,79,180,130,240)(5,124,37,205,80,181,131,241)(6,94,38,206,81,182,132,242)(7,95,39,207,82,183,133,243)(8,96,40,208,83,184,134,244)(9,97,41,209,84,185,135,245)(10,98,42,210,85,186,136,246)(11,99,43,211,86,156,137,247)(12,100,44,212,87,157,138,248)(13,101,45,213,88,158,139,218)(14,102,46,214,89,159,140,219)(15,103,47,215,90,160,141,220)(16,104,48,216,91,161,142,221)(17,105,49,217,92,162,143,222)(18,106,50,187,93,163,144,223)(19,107,51,188,63,164,145,224)(20,108,52,189,64,165,146,225)(21,109,53,190,65,166,147,226)(22,110,54,191,66,167,148,227)(23,111,55,192,67,168,149,228)(24,112,56,193,68,169,150,229)(25,113,57,194,69,170,151,230)(26,114,58,195,70,171,152,231)(27,115,59,196,71,172,153,232)(28,116,60,197,72,173,154,233)(29,117,61,198,73,174,155,234)(30,118,62,199,74,175,125,235)(31,119,32,200,75,176,126,236), (32,126)(33,127)(34,128)(35,129)(36,130)(37,131)(38,132)(39,133)(40,134)(41,135)(42,136)(43,137)(44,138)(45,139)(46,140)(47,141)(48,142)(49,143)(50,144)(51,145)(52,146)(53,147)(54,148)(55,149)(56,150)(57,151)(58,152)(59,153)(60,154)(61,155)(62,125)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,187)(107,188)(108,189)(109,190)(110,191)(111,192)(112,193)(113,194)(114,195)(115,196)(116,197)(117,198)(118,199)(119,200)(120,201)(121,202)(122,203)(123,204)(124,205)(156,247)(157,248)(158,218)(159,219)(160,220)(161,221)(162,222)(163,223)(164,224)(165,225)(166,226)(167,227)(168,228)(169,229)(170,230)(171,231)(172,232)(173,233)(174,234)(175,235)(176,236)(177,237)(178,238)(179,239)(180,240)(181,241)(182,242)(183,243)(184,244)(185,245)(186,246)>;
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)(218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248), (1,120,33,201,76,177,127,237)(2,121,34,202,77,178,128,238)(3,122,35,203,78,179,129,239)(4,123,36,204,79,180,130,240)(5,124,37,205,80,181,131,241)(6,94,38,206,81,182,132,242)(7,95,39,207,82,183,133,243)(8,96,40,208,83,184,134,244)(9,97,41,209,84,185,135,245)(10,98,42,210,85,186,136,246)(11,99,43,211,86,156,137,247)(12,100,44,212,87,157,138,248)(13,101,45,213,88,158,139,218)(14,102,46,214,89,159,140,219)(15,103,47,215,90,160,141,220)(16,104,48,216,91,161,142,221)(17,105,49,217,92,162,143,222)(18,106,50,187,93,163,144,223)(19,107,51,188,63,164,145,224)(20,108,52,189,64,165,146,225)(21,109,53,190,65,166,147,226)(22,110,54,191,66,167,148,227)(23,111,55,192,67,168,149,228)(24,112,56,193,68,169,150,229)(25,113,57,194,69,170,151,230)(26,114,58,195,70,171,152,231)(27,115,59,196,71,172,153,232)(28,116,60,197,72,173,154,233)(29,117,61,198,73,174,155,234)(30,118,62,199,74,175,125,235)(31,119,32,200,75,176,126,236), (32,126)(33,127)(34,128)(35,129)(36,130)(37,131)(38,132)(39,133)(40,134)(41,135)(42,136)(43,137)(44,138)(45,139)(46,140)(47,141)(48,142)(49,143)(50,144)(51,145)(52,146)(53,147)(54,148)(55,149)(56,150)(57,151)(58,152)(59,153)(60,154)(61,155)(62,125)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,187)(107,188)(108,189)(109,190)(110,191)(111,192)(112,193)(113,194)(114,195)(115,196)(116,197)(117,198)(118,199)(119,200)(120,201)(121,202)(122,203)(123,204)(124,205)(156,247)(157,248)(158,218)(159,219)(160,220)(161,221)(162,222)(163,223)(164,224)(165,225)(166,226)(167,227)(168,228)(169,229)(170,230)(171,231)(172,232)(173,233)(174,234)(175,235)(176,236)(177,237)(178,238)(179,239)(180,240)(181,241)(182,242)(183,243)(184,244)(185,245)(186,246) );
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),(218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248)], [(1,120,33,201,76,177,127,237),(2,121,34,202,77,178,128,238),(3,122,35,203,78,179,129,239),(4,123,36,204,79,180,130,240),(5,124,37,205,80,181,131,241),(6,94,38,206,81,182,132,242),(7,95,39,207,82,183,133,243),(8,96,40,208,83,184,134,244),(9,97,41,209,84,185,135,245),(10,98,42,210,85,186,136,246),(11,99,43,211,86,156,137,247),(12,100,44,212,87,157,138,248),(13,101,45,213,88,158,139,218),(14,102,46,214,89,159,140,219),(15,103,47,215,90,160,141,220),(16,104,48,216,91,161,142,221),(17,105,49,217,92,162,143,222),(18,106,50,187,93,163,144,223),(19,107,51,188,63,164,145,224),(20,108,52,189,64,165,146,225),(21,109,53,190,65,166,147,226),(22,110,54,191,66,167,148,227),(23,111,55,192,67,168,149,228),(24,112,56,193,68,169,150,229),(25,113,57,194,69,170,151,230),(26,114,58,195,70,171,152,231),(27,115,59,196,71,172,153,232),(28,116,60,197,72,173,154,233),(29,117,61,198,73,174,155,234),(30,118,62,199,74,175,125,235),(31,119,32,200,75,176,126,236)], [(32,126),(33,127),(34,128),(35,129),(36,130),(37,131),(38,132),(39,133),(40,134),(41,135),(42,136),(43,137),(44,138),(45,139),(46,140),(47,141),(48,142),(49,143),(50,144),(51,145),(52,146),(53,147),(54,148),(55,149),(56,150),(57,151),(58,152),(59,153),(60,154),(61,155),(62,125),(94,206),(95,207),(96,208),(97,209),(98,210),(99,211),(100,212),(101,213),(102,214),(103,215),(104,216),(105,217),(106,187),(107,188),(108,189),(109,190),(110,191),(111,192),(112,193),(113,194),(114,195),(115,196),(116,197),(117,198),(118,199),(119,200),(120,201),(121,202),(122,203),(123,204),(124,205),(156,247),(157,248),(158,218),(159,219),(160,220),(161,221),(162,222),(163,223),(164,224),(165,225),(166,226),(167,227),(168,228),(169,229),(170,230),(171,231),(172,232),(173,233),(174,234),(175,235),(176,236),(177,237),(178,238),(179,239),(180,240),(181,241),(182,242),(183,243),(184,244),(185,245),(186,246)]])
217 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 31A | ··· | 31AD | 62A | ··· | 62AD | 62AE | ··· | 62BH | 124A | ··· | 124AD | 124AE | ··· | 124BH | 248A | ··· | 248BH |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 31 | ··· | 31 | 62 | ··· | 62 | 62 | ··· | 62 | 124 | ··· | 124 | 124 | ··· | 124 | 248 | ··· | 248 |
| size | 1 | 1 | 4 | 2 | 4 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
217 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C31 | C62 | C62 | C62 | D4 | SD16 | D4×C31 | SD16×C31 |
| kernel | SD16×C31 | C248 | D4×C31 | Q8×C31 | SD16 | C8 | D4 | Q8 | C62 | C31 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 30 | 30 | 30 | 30 | 1 | 2 | 30 | 60 |
Matrix representation of SD16×C31 ►in GL2(𝔽1489) generated by
| 296 | 0 |
| 0 | 296 |
| 206 | 1283 |
| 206 | 206 |
| 1 | 0 |
| 0 | 1488 |
G:=sub<GL(2,GF(1489))| [296,0,0,296],[206,206,1283,206],[1,0,0,1488] >;
SD16×C31 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\times C_{31} % in TeX
G:=Group("SD16xC31"); // GroupNames label
G:=SmallGroup(496,25);
// by ID
G=gap.SmallGroup(496,25);
# by ID
G:=PCGroup([5,-2,-2,-31,-2,-2,1240,1261,7443,3728,58]);
// Polycyclic
G:=Group<a,b,c|a^31=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations
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