direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: D8×C31, D4⋊C62, C8⋊1C62, C248⋊5C2, C62.14D4, C124.17C22, (D4×C31)⋊4C2, C4.1(C2×C62), C2.3(D4×C31), SmallGroup(496,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8×C31
G = < a,b,c | a31=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D8×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)
(1 246 98 214 56 171 137 65)(2 247 99 215 57 172 138 66)(3 248 100 216 58 173 139 67)(4 218 101 217 59 174 140 68)(5 219 102 187 60 175 141 69)(6 220 103 188 61 176 142 70)(7 221 104 189 62 177 143 71)(8 222 105 190 32 178 144 72)(9 223 106 191 33 179 145 73)(10 224 107 192 34 180 146 74)(11 225 108 193 35 181 147 75)(12 226 109 194 36 182 148 76)(13 227 110 195 37 183 149 77)(14 228 111 196 38 184 150 78)(15 229 112 197 39 185 151 79)(16 230 113 198 40 186 152 80)(17 231 114 199 41 156 153 81)(18 232 115 200 42 157 154 82)(19 233 116 201 43 158 155 83)(20 234 117 202 44 159 125 84)(21 235 118 203 45 160 126 85)(22 236 119 204 46 161 127 86)(23 237 120 205 47 162 128 87)(24 238 121 206 48 163 129 88)(25 239 122 207 49 164 130 89)(26 240 123 208 50 165 131 90)(27 241 124 209 51 166 132 91)(28 242 94 210 52 167 133 92)(29 243 95 211 53 168 134 93)(30 244 96 212 54 169 135 63)(31 245 97 213 55 170 136 64)
(63 244)(64 245)(65 246)(66 247)(67 248)(68 218)(69 219)(70 220)(71 221)(72 222)(73 223)(74 224)(75 225)(76 226)(77 227)(78 228)(79 229)(80 230)(81 231)(82 232)(83 233)(84 234)(85 235)(86 236)(87 237)(88 238)(89 239)(90 240)(91 241)(92 242)(93 243)(94 133)(95 134)(96 135)(97 136)(98 137)(99 138)(100 139)(101 140)(102 141)(103 142)(104 143)(105 144)(106 145)(107 146)(108 147)(109 148)(110 149)(111 150)(112 151)(113 152)(114 153)(115 154)(116 155)(117 125)(118 126)(119 127)(120 128)(121 129)(122 130)(123 131)(124 132)(156 199)(157 200)(158 201)(159 202)(160 203)(161 204)(162 205)(163 206)(164 207)(165 208)(166 209)(167 210)(168 211)(169 212)(170 213)(171 214)(172 215)(173 216)(174 217)(175 187)(176 188)(177 189)(178 190)(179 191)(180 192)(181 193)(182 194)(183 195)(184 196)(185 197)(186 198)
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), (1,246,98,214,56,171,137,65)(2,247,99,215,57,172,138,66)(3,248,100,216,58,173,139,67)(4,218,101,217,59,174,140,68)(5,219,102,187,60,175,141,69)(6,220,103,188,61,176,142,70)(7,221,104,189,62,177,143,71)(8,222,105,190,32,178,144,72)(9,223,106,191,33,179,145,73)(10,224,107,192,34,180,146,74)(11,225,108,193,35,181,147,75)(12,226,109,194,36,182,148,76)(13,227,110,195,37,183,149,77)(14,228,111,196,38,184,150,78)(15,229,112,197,39,185,151,79)(16,230,113,198,40,186,152,80)(17,231,114,199,41,156,153,81)(18,232,115,200,42,157,154,82)(19,233,116,201,43,158,155,83)(20,234,117,202,44,159,125,84)(21,235,118,203,45,160,126,85)(22,236,119,204,46,161,127,86)(23,237,120,205,47,162,128,87)(24,238,121,206,48,163,129,88)(25,239,122,207,49,164,130,89)(26,240,123,208,50,165,131,90)(27,241,124,209,51,166,132,91)(28,242,94,210,52,167,133,92)(29,243,95,211,53,168,134,93)(30,244,96,212,54,169,135,63)(31,245,97,213,55,170,136,64), (63,244)(64,245)(65,246)(66,247)(67,248)(68,218)(69,219)(70,220)(71,221)(72,222)(73,223)(74,224)(75,225)(76,226)(77,227)(78,228)(79,229)(80,230)(81,231)(82,232)(83,233)(84,234)(85,235)(86,236)(87,237)(88,238)(89,239)(90,240)(91,241)(92,242)(93,243)(94,133)(95,134)(96,135)(97,136)(98,137)(99,138)(100,139)(101,140)(102,141)(103,142)(104,143)(105,144)(106,145)(107,146)(108,147)(109,148)(110,149)(111,150)(112,151)(113,152)(114,153)(115,154)(116,155)(117,125)(118,126)(119,127)(120,128)(121,129)(122,130)(123,131)(124,132)(156,199)(157,200)(158,201)(159,202)(160,203)(161,204)(162,205)(163,206)(164,207)(165,208)(166,209)(167,210)(168,211)(169,212)(170,213)(171,214)(172,215)(173,216)(174,217)(175,187)(176,188)(177,189)(178,190)(179,191)(180,192)(181,193)(182,194)(183,195)(184,196)(185,197)(186,198)>;
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), (1,246,98,214,56,171,137,65)(2,247,99,215,57,172,138,66)(3,248,100,216,58,173,139,67)(4,218,101,217,59,174,140,68)(5,219,102,187,60,175,141,69)(6,220,103,188,61,176,142,70)(7,221,104,189,62,177,143,71)(8,222,105,190,32,178,144,72)(9,223,106,191,33,179,145,73)(10,224,107,192,34,180,146,74)(11,225,108,193,35,181,147,75)(12,226,109,194,36,182,148,76)(13,227,110,195,37,183,149,77)(14,228,111,196,38,184,150,78)(15,229,112,197,39,185,151,79)(16,230,113,198,40,186,152,80)(17,231,114,199,41,156,153,81)(18,232,115,200,42,157,154,82)(19,233,116,201,43,158,155,83)(20,234,117,202,44,159,125,84)(21,235,118,203,45,160,126,85)(22,236,119,204,46,161,127,86)(23,237,120,205,47,162,128,87)(24,238,121,206,48,163,129,88)(25,239,122,207,49,164,130,89)(26,240,123,208,50,165,131,90)(27,241,124,209,51,166,132,91)(28,242,94,210,52,167,133,92)(29,243,95,211,53,168,134,93)(30,244,96,212,54,169,135,63)(31,245,97,213,55,170,136,64), (63,244)(64,245)(65,246)(66,247)(67,248)(68,218)(69,219)(70,220)(71,221)(72,222)(73,223)(74,224)(75,225)(76,226)(77,227)(78,228)(79,229)(80,230)(81,231)(82,232)(83,233)(84,234)(85,235)(86,236)(87,237)(88,238)(89,239)(90,240)(91,241)(92,242)(93,243)(94,133)(95,134)(96,135)(97,136)(98,137)(99,138)(100,139)(101,140)(102,141)(103,142)(104,143)(105,144)(106,145)(107,146)(108,147)(109,148)(110,149)(111,150)(112,151)(113,152)(114,153)(115,154)(116,155)(117,125)(118,126)(119,127)(120,128)(121,129)(122,130)(123,131)(124,132)(156,199)(157,200)(158,201)(159,202)(160,203)(161,204)(162,205)(163,206)(164,207)(165,208)(166,209)(167,210)(168,211)(169,212)(170,213)(171,214)(172,215)(173,216)(174,217)(175,187)(176,188)(177,189)(178,190)(179,191)(180,192)(181,193)(182,194)(183,195)(184,196)(185,197)(186,198) );
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)], [(1,246,98,214,56,171,137,65),(2,247,99,215,57,172,138,66),(3,248,100,216,58,173,139,67),(4,218,101,217,59,174,140,68),(5,219,102,187,60,175,141,69),(6,220,103,188,61,176,142,70),(7,221,104,189,62,177,143,71),(8,222,105,190,32,178,144,72),(9,223,106,191,33,179,145,73),(10,224,107,192,34,180,146,74),(11,225,108,193,35,181,147,75),(12,226,109,194,36,182,148,76),(13,227,110,195,37,183,149,77),(14,228,111,196,38,184,150,78),(15,229,112,197,39,185,151,79),(16,230,113,198,40,186,152,80),(17,231,114,199,41,156,153,81),(18,232,115,200,42,157,154,82),(19,233,116,201,43,158,155,83),(20,234,117,202,44,159,125,84),(21,235,118,203,45,160,126,85),(22,236,119,204,46,161,127,86),(23,237,120,205,47,162,128,87),(24,238,121,206,48,163,129,88),(25,239,122,207,49,164,130,89),(26,240,123,208,50,165,131,90),(27,241,124,209,51,166,132,91),(28,242,94,210,52,167,133,92),(29,243,95,211,53,168,134,93),(30,244,96,212,54,169,135,63),(31,245,97,213,55,170,136,64)], [(63,244),(64,245),(65,246),(66,247),(67,248),(68,218),(69,219),(70,220),(71,221),(72,222),(73,223),(74,224),(75,225),(76,226),(77,227),(78,228),(79,229),(80,230),(81,231),(82,232),(83,233),(84,234),(85,235),(86,236),(87,237),(88,238),(89,239),(90,240),(91,241),(92,242),(93,243),(94,133),(95,134),(96,135),(97,136),(98,137),(99,138),(100,139),(101,140),(102,141),(103,142),(104,143),(105,144),(106,145),(107,146),(108,147),(109,148),(110,149),(111,150),(112,151),(113,152),(114,153),(115,154),(116,155),(117,125),(118,126),(119,127),(120,128),(121,129),(122,130),(123,131),(124,132),(156,199),(157,200),(158,201),(159,202),(160,203),(161,204),(162,205),(163,206),(164,207),(165,208),(166,209),(167,210),(168,211),(169,212),(170,213),(171,214),(172,215),(173,216),(174,217),(175,187),(176,188),(177,189),(178,190),(179,191),(180,192),(181,193),(182,194),(183,195),(184,196),(185,197),(186,198)]])
217 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 8A | 8B | 31A | ··· | 31AD | 62A | ··· | 62AD | 62AE | ··· | 62CL | 124A | ··· | 124AD | 248A | ··· | 248BH |
| order | 1 | 2 | 2 | 2 | 4 | 8 | 8 | 31 | ··· | 31 | 62 | ··· | 62 | 62 | ··· | 62 | 124 | ··· | 124 | 248 | ··· | 248 |
| size | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 |
217 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C31 | C62 | C62 | D4 | D8 | D4×C31 | D8×C31 |
| kernel | D8×C31 | C248 | D4×C31 | D8 | C8 | D4 | C62 | C31 | C2 | C1 |
| # reps | 1 | 1 | 2 | 30 | 30 | 60 | 1 | 2 | 30 | 60 |
Matrix representation of D8×C31 ►in GL2(𝔽1489) generated by
| 1254 | 0 |
| 0 | 1254 |
| 191 | 1298 |
| 191 | 191 |
| 1 | 0 |
| 0 | 1488 |
G:=sub<GL(2,GF(1489))| [1254,0,0,1254],[191,191,1298,191],[1,0,0,1488] >;
D8×C31 in GAP, Magma, Sage, TeX
D_8\times C_{31} % in TeX
G:=Group("D8xC31"); // GroupNames label
G:=SmallGroup(496,24);
// by ID
G=gap.SmallGroup(496,24);
# by ID
G:=PCGroup([5,-2,-2,-31,-2,-2,1261,7443,3728,58]);
// 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^-1>;
// generators/relations
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