direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C8×D31, C248⋊3C2, D62.2C4, C4.12D62, Dic31.2C4, C124.12C22, C31⋊C8⋊6C2, C31⋊1(C2×C8), C62.1(C2×C4), C2.1(C4×D31), (C4×D31).3C2, SmallGroup(496,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C31 — C8×D31 |
Generators and relations for C8×D31
G = < a,b,c | a8=b31=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C8×D31
►On 248 points
Generators in S248
(1 238 111 174 56 210 65 136)(2 239 112 175 57 211 66 137)(3 240 113 176 58 212 67 138)(4 241 114 177 59 213 68 139)(5 242 115 178 60 214 69 140)(6 243 116 179 61 215 70 141)(7 244 117 180 62 216 71 142)(8 245 118 181 32 217 72 143)(9 246 119 182 33 187 73 144)(10 247 120 183 34 188 74 145)(11 248 121 184 35 189 75 146)(12 218 122 185 36 190 76 147)(13 219 123 186 37 191 77 148)(14 220 124 156 38 192 78 149)(15 221 94 157 39 193 79 150)(16 222 95 158 40 194 80 151)(17 223 96 159 41 195 81 152)(18 224 97 160 42 196 82 153)(19 225 98 161 43 197 83 154)(20 226 99 162 44 198 84 155)(21 227 100 163 45 199 85 125)(22 228 101 164 46 200 86 126)(23 229 102 165 47 201 87 127)(24 230 103 166 48 202 88 128)(25 231 104 167 49 203 89 129)(26 232 105 168 50 204 90 130)(27 233 106 169 51 205 91 131)(28 234 107 170 52 206 92 132)(29 235 108 171 53 207 93 133)(30 236 109 172 54 208 63 134)(31 237 110 173 55 209 64 135)
(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 55)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 48)(9 47)(10 46)(11 45)(12 44)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 62)(26 61)(27 60)(28 59)(29 58)(30 57)(31 56)(63 112)(64 111)(65 110)(66 109)(67 108)(68 107)(69 106)(70 105)(71 104)(72 103)(73 102)(74 101)(75 100)(76 99)(77 98)(78 97)(79 96)(80 95)(81 94)(82 124)(83 123)(84 122)(85 121)(86 120)(87 119)(88 118)(89 117)(90 116)(91 115)(92 114)(93 113)(125 184)(126 183)(127 182)(128 181)(129 180)(130 179)(131 178)(132 177)(133 176)(134 175)(135 174)(136 173)(137 172)(138 171)(139 170)(140 169)(141 168)(142 167)(143 166)(144 165)(145 164)(146 163)(147 162)(148 161)(149 160)(150 159)(151 158)(152 157)(153 156)(154 186)(155 185)(187 229)(188 228)(189 227)(190 226)(191 225)(192 224)(193 223)(194 222)(195 221)(196 220)(197 219)(198 218)(199 248)(200 247)(201 246)(202 245)(203 244)(204 243)(205 242)(206 241)(207 240)(208 239)(209 238)(210 237)(211 236)(212 235)(213 234)(214 233)(215 232)(216 231)(217 230)
G:=sub<Sym(248)| (1,238,111,174,56,210,65,136)(2,239,112,175,57,211,66,137)(3,240,113,176,58,212,67,138)(4,241,114,177,59,213,68,139)(5,242,115,178,60,214,69,140)(6,243,116,179,61,215,70,141)(7,244,117,180,62,216,71,142)(8,245,118,181,32,217,72,143)(9,246,119,182,33,187,73,144)(10,247,120,183,34,188,74,145)(11,248,121,184,35,189,75,146)(12,218,122,185,36,190,76,147)(13,219,123,186,37,191,77,148)(14,220,124,156,38,192,78,149)(15,221,94,157,39,193,79,150)(16,222,95,158,40,194,80,151)(17,223,96,159,41,195,81,152)(18,224,97,160,42,196,82,153)(19,225,98,161,43,197,83,154)(20,226,99,162,44,198,84,155)(21,227,100,163,45,199,85,125)(22,228,101,164,46,200,86,126)(23,229,102,165,47,201,87,127)(24,230,103,166,48,202,88,128)(25,231,104,167,49,203,89,129)(26,232,105,168,50,204,90,130)(27,233,106,169,51,205,91,131)(28,234,107,170,52,206,92,132)(29,235,108,171,53,207,93,133)(30,236,109,172,54,208,63,134)(31,237,110,173,55,209,64,135), (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), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,62)(26,61)(27,60)(28,59)(29,58)(30,57)(31,56)(63,112)(64,111)(65,110)(66,109)(67,108)(68,107)(69,106)(70,105)(71,104)(72,103)(73,102)(74,101)(75,100)(76,99)(77,98)(78,97)(79,96)(80,95)(81,94)(82,124)(83,123)(84,122)(85,121)(86,120)(87,119)(88,118)(89,117)(90,116)(91,115)(92,114)(93,113)(125,184)(126,183)(127,182)(128,181)(129,180)(130,179)(131,178)(132,177)(133,176)(134,175)(135,174)(136,173)(137,172)(138,171)(139,170)(140,169)(141,168)(142,167)(143,166)(144,165)(145,164)(146,163)(147,162)(148,161)(149,160)(150,159)(151,158)(152,157)(153,156)(154,186)(155,185)(187,229)(188,228)(189,227)(190,226)(191,225)(192,224)(193,223)(194,222)(195,221)(196,220)(197,219)(198,218)(199,248)(200,247)(201,246)(202,245)(203,244)(204,243)(205,242)(206,241)(207,240)(208,239)(209,238)(210,237)(211,236)(212,235)(213,234)(214,233)(215,232)(216,231)(217,230)>;
G:=Group( (1,238,111,174,56,210,65,136)(2,239,112,175,57,211,66,137)(3,240,113,176,58,212,67,138)(4,241,114,177,59,213,68,139)(5,242,115,178,60,214,69,140)(6,243,116,179,61,215,70,141)(7,244,117,180,62,216,71,142)(8,245,118,181,32,217,72,143)(9,246,119,182,33,187,73,144)(10,247,120,183,34,188,74,145)(11,248,121,184,35,189,75,146)(12,218,122,185,36,190,76,147)(13,219,123,186,37,191,77,148)(14,220,124,156,38,192,78,149)(15,221,94,157,39,193,79,150)(16,222,95,158,40,194,80,151)(17,223,96,159,41,195,81,152)(18,224,97,160,42,196,82,153)(19,225,98,161,43,197,83,154)(20,226,99,162,44,198,84,155)(21,227,100,163,45,199,85,125)(22,228,101,164,46,200,86,126)(23,229,102,165,47,201,87,127)(24,230,103,166,48,202,88,128)(25,231,104,167,49,203,89,129)(26,232,105,168,50,204,90,130)(27,233,106,169,51,205,91,131)(28,234,107,170,52,206,92,132)(29,235,108,171,53,207,93,133)(30,236,109,172,54,208,63,134)(31,237,110,173,55,209,64,135), (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), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,62)(26,61)(27,60)(28,59)(29,58)(30,57)(31,56)(63,112)(64,111)(65,110)(66,109)(67,108)(68,107)(69,106)(70,105)(71,104)(72,103)(73,102)(74,101)(75,100)(76,99)(77,98)(78,97)(79,96)(80,95)(81,94)(82,124)(83,123)(84,122)(85,121)(86,120)(87,119)(88,118)(89,117)(90,116)(91,115)(92,114)(93,113)(125,184)(126,183)(127,182)(128,181)(129,180)(130,179)(131,178)(132,177)(133,176)(134,175)(135,174)(136,173)(137,172)(138,171)(139,170)(140,169)(141,168)(142,167)(143,166)(144,165)(145,164)(146,163)(147,162)(148,161)(149,160)(150,159)(151,158)(152,157)(153,156)(154,186)(155,185)(187,229)(188,228)(189,227)(190,226)(191,225)(192,224)(193,223)(194,222)(195,221)(196,220)(197,219)(198,218)(199,248)(200,247)(201,246)(202,245)(203,244)(204,243)(205,242)(206,241)(207,240)(208,239)(209,238)(210,237)(211,236)(212,235)(213,234)(214,233)(215,232)(216,231)(217,230) );
G=PermutationGroup([[(1,238,111,174,56,210,65,136),(2,239,112,175,57,211,66,137),(3,240,113,176,58,212,67,138),(4,241,114,177,59,213,68,139),(5,242,115,178,60,214,69,140),(6,243,116,179,61,215,70,141),(7,244,117,180,62,216,71,142),(8,245,118,181,32,217,72,143),(9,246,119,182,33,187,73,144),(10,247,120,183,34,188,74,145),(11,248,121,184,35,189,75,146),(12,218,122,185,36,190,76,147),(13,219,123,186,37,191,77,148),(14,220,124,156,38,192,78,149),(15,221,94,157,39,193,79,150),(16,222,95,158,40,194,80,151),(17,223,96,159,41,195,81,152),(18,224,97,160,42,196,82,153),(19,225,98,161,43,197,83,154),(20,226,99,162,44,198,84,155),(21,227,100,163,45,199,85,125),(22,228,101,164,46,200,86,126),(23,229,102,165,47,201,87,127),(24,230,103,166,48,202,88,128),(25,231,104,167,49,203,89,129),(26,232,105,168,50,204,90,130),(27,233,106,169,51,205,91,131),(28,234,107,170,52,206,92,132),(29,235,108,171,53,207,93,133),(30,236,109,172,54,208,63,134),(31,237,110,173,55,209,64,135)], [(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)], [(1,55),(2,54),(3,53),(4,52),(5,51),(6,50),(7,49),(8,48),(9,47),(10,46),(11,45),(12,44),(13,43),(14,42),(15,41),(16,40),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(24,32),(25,62),(26,61),(27,60),(28,59),(29,58),(30,57),(31,56),(63,112),(64,111),(65,110),(66,109),(67,108),(68,107),(69,106),(70,105),(71,104),(72,103),(73,102),(74,101),(75,100),(76,99),(77,98),(78,97),(79,96),(80,95),(81,94),(82,124),(83,123),(84,122),(85,121),(86,120),(87,119),(88,118),(89,117),(90,116),(91,115),(92,114),(93,113),(125,184),(126,183),(127,182),(128,181),(129,180),(130,179),(131,178),(132,177),(133,176),(134,175),(135,174),(136,173),(137,172),(138,171),(139,170),(140,169),(141,168),(142,167),(143,166),(144,165),(145,164),(146,163),(147,162),(148,161),(149,160),(150,159),(151,158),(152,157),(153,156),(154,186),(155,185),(187,229),(188,228),(189,227),(190,226),(191,225),(192,224),(193,223),(194,222),(195,221),(196,220),(197,219),(198,218),(199,248),(200,247),(201,246),(202,245),(203,244),(204,243),(205,242),(206,241),(207,240),(208,239),(209,238),(210,237),(211,236),(212,235),(213,234),(214,233),(215,232),(216,231),(217,230)]])
136 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 31A | ··· | 31O | 62A | ··· | 62O | 124A | ··· | 124AD | 248A | ··· | 248BH |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 | 248 | ··· | 248 |
| size | 1 | 1 | 31 | 31 | 1 | 1 | 31 | 31 | 1 | 1 | 1 | 1 | 31 | 31 | 31 | 31 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D31 | D62 | C4×D31 | C8×D31 |
| kernel | C8×D31 | C31⋊C8 | C248 | C4×D31 | Dic31 | D62 | D31 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 15 | 15 | 30 | 60 |
Matrix representation of C8×D31 ►in GL3(𝔽1489) generated by
| 15 | 0 | 0 |
| 0 | 1264 | 0 |
| 0 | 0 | 1264 |
| 1 | 0 | 0 |
| 0 | 1211 | 1 |
| 0 | 1398 | 943 |
| 1 | 0 | 0 |
| 0 | 1163 | 213 |
| 0 | 1032 | 326 |
G:=sub<GL(3,GF(1489))| [15,0,0,0,1264,0,0,0,1264],[1,0,0,0,1211,1398,0,1,943],[1,0,0,0,1163,1032,0,213,326] >;
C8×D31 in GAP, Magma, Sage, TeX
C_8\times D_{31} % in TeX
G:=Group("C8xD31"); // GroupNames label
G:=SmallGroup(496,3);
// by ID
G=gap.SmallGroup(496,3);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-31,26,42,12004]);
// Polycyclic
G:=Group<a,b,c|a^8=b^31=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export