metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊3D31, D62.C4, C248⋊4C2, C4.13D62, Dic31.C4, C31⋊1M4(2), C124.13C22, C31⋊C8⋊4C2, C62.2(C2×C4), C2.3(C4×D31), (C4×D31).2C2, SmallGroup(496,4)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊D31
G = < a,b,c | a8=b31=c2=1, ab=ba, cac=a5, cbc=b-1 >
Smallest permutation representation of C8⋊D31
►On 248 points
Generators in S248
(1 232 98 183 55 215 65 136)(2 233 99 184 56 216 66 137)(3 234 100 185 57 217 67 138)(4 235 101 186 58 187 68 139)(5 236 102 156 59 188 69 140)(6 237 103 157 60 189 70 141)(7 238 104 158 61 190 71 142)(8 239 105 159 62 191 72 143)(9 240 106 160 32 192 73 144)(10 241 107 161 33 193 74 145)(11 242 108 162 34 194 75 146)(12 243 109 163 35 195 76 147)(13 244 110 164 36 196 77 148)(14 245 111 165 37 197 78 149)(15 246 112 166 38 198 79 150)(16 247 113 167 39 199 80 151)(17 248 114 168 40 200 81 152)(18 218 115 169 41 201 82 153)(19 219 116 170 42 202 83 154)(20 220 117 171 43 203 84 155)(21 221 118 172 44 204 85 125)(22 222 119 173 45 205 86 126)(23 223 120 174 46 206 87 127)(24 224 121 175 47 207 88 128)(25 225 122 176 48 208 89 129)(26 226 123 177 49 209 90 130)(27 227 124 178 50 210 91 131)(28 228 94 179 51 211 92 132)(29 229 95 180 52 212 93 133)(30 230 96 181 53 213 63 134)(31 231 97 182 54 214 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 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 21)(12 20)(13 19)(14 18)(15 17)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(47 62)(48 61)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(63 66)(64 65)(67 93)(68 92)(69 91)(70 90)(71 89)(72 88)(73 87)(74 86)(75 85)(76 84)(77 83)(78 82)(79 81)(94 101)(95 100)(96 99)(97 98)(102 124)(103 123)(104 122)(105 121)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)(125 162)(126 161)(127 160)(128 159)(129 158)(130 157)(131 156)(132 186)(133 185)(134 184)(135 183)(136 182)(137 181)(138 180)(139 179)(140 178)(141 177)(142 176)(143 175)(144 174)(145 173)(146 172)(147 171)(148 170)(149 169)(150 168)(151 167)(152 166)(153 165)(154 164)(155 163)(187 228)(188 227)(189 226)(190 225)(191 224)(192 223)(193 222)(194 221)(195 220)(196 219)(197 218)(198 248)(199 247)(200 246)(201 245)(202 244)(203 243)(204 242)(205 241)(206 240)(207 239)(208 238)(209 237)(210 236)(211 235)(212 234)(213 233)(214 232)(215 231)(216 230)(217 229)
G:=sub<Sym(248)| (1,232,98,183,55,215,65,136)(2,233,99,184,56,216,66,137)(3,234,100,185,57,217,67,138)(4,235,101,186,58,187,68,139)(5,236,102,156,59,188,69,140)(6,237,103,157,60,189,70,141)(7,238,104,158,61,190,71,142)(8,239,105,159,62,191,72,143)(9,240,106,160,32,192,73,144)(10,241,107,161,33,193,74,145)(11,242,108,162,34,194,75,146)(12,243,109,163,35,195,76,147)(13,244,110,164,36,196,77,148)(14,245,111,165,37,197,78,149)(15,246,112,166,38,198,79,150)(16,247,113,167,39,199,80,151)(17,248,114,168,40,200,81,152)(18,218,115,169,41,201,82,153)(19,219,116,170,42,202,83,154)(20,220,117,171,43,203,84,155)(21,221,118,172,44,204,85,125)(22,222,119,173,45,205,86,126)(23,223,120,174,46,206,87,127)(24,224,121,175,47,207,88,128)(25,225,122,176,48,208,89,129)(26,226,123,177,49,209,90,130)(27,227,124,178,50,210,91,131)(28,228,94,179,51,211,92,132)(29,229,95,180,52,212,93,133)(30,230,96,181,53,213,63,134)(31,231,97,182,54,214,64,135), (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), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(63,66)(64,65)(67,93)(68,92)(69,91)(70,90)(71,89)(72,88)(73,87)(74,86)(75,85)(76,84)(77,83)(78,82)(79,81)(94,101)(95,100)(96,99)(97,98)(102,124)(103,123)(104,122)(105,121)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114)(125,162)(126,161)(127,160)(128,159)(129,158)(130,157)(131,156)(132,186)(133,185)(134,184)(135,183)(136,182)(137,181)(138,180)(139,179)(140,178)(141,177)(142,176)(143,175)(144,174)(145,173)(146,172)(147,171)(148,170)(149,169)(150,168)(151,167)(152,166)(153,165)(154,164)(155,163)(187,228)(188,227)(189,226)(190,225)(191,224)(192,223)(193,222)(194,221)(195,220)(196,219)(197,218)(198,248)(199,247)(200,246)(201,245)(202,244)(203,243)(204,242)(205,241)(206,240)(207,239)(208,238)(209,237)(210,236)(211,235)(212,234)(213,233)(214,232)(215,231)(216,230)(217,229)>;
G:=Group( (1,232,98,183,55,215,65,136)(2,233,99,184,56,216,66,137)(3,234,100,185,57,217,67,138)(4,235,101,186,58,187,68,139)(5,236,102,156,59,188,69,140)(6,237,103,157,60,189,70,141)(7,238,104,158,61,190,71,142)(8,239,105,159,62,191,72,143)(9,240,106,160,32,192,73,144)(10,241,107,161,33,193,74,145)(11,242,108,162,34,194,75,146)(12,243,109,163,35,195,76,147)(13,244,110,164,36,196,77,148)(14,245,111,165,37,197,78,149)(15,246,112,166,38,198,79,150)(16,247,113,167,39,199,80,151)(17,248,114,168,40,200,81,152)(18,218,115,169,41,201,82,153)(19,219,116,170,42,202,83,154)(20,220,117,171,43,203,84,155)(21,221,118,172,44,204,85,125)(22,222,119,173,45,205,86,126)(23,223,120,174,46,206,87,127)(24,224,121,175,47,207,88,128)(25,225,122,176,48,208,89,129)(26,226,123,177,49,209,90,130)(27,227,124,178,50,210,91,131)(28,228,94,179,51,211,92,132)(29,229,95,180,52,212,93,133)(30,230,96,181,53,213,63,134)(31,231,97,182,54,214,64,135), (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), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(63,66)(64,65)(67,93)(68,92)(69,91)(70,90)(71,89)(72,88)(73,87)(74,86)(75,85)(76,84)(77,83)(78,82)(79,81)(94,101)(95,100)(96,99)(97,98)(102,124)(103,123)(104,122)(105,121)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114)(125,162)(126,161)(127,160)(128,159)(129,158)(130,157)(131,156)(132,186)(133,185)(134,184)(135,183)(136,182)(137,181)(138,180)(139,179)(140,178)(141,177)(142,176)(143,175)(144,174)(145,173)(146,172)(147,171)(148,170)(149,169)(150,168)(151,167)(152,166)(153,165)(154,164)(155,163)(187,228)(188,227)(189,226)(190,225)(191,224)(192,223)(193,222)(194,221)(195,220)(196,219)(197,218)(198,248)(199,247)(200,246)(201,245)(202,244)(203,243)(204,242)(205,241)(206,240)(207,239)(208,238)(209,237)(210,236)(211,235)(212,234)(213,233)(214,232)(215,231)(216,230)(217,229) );
G=PermutationGroup([[(1,232,98,183,55,215,65,136),(2,233,99,184,56,216,66,137),(3,234,100,185,57,217,67,138),(4,235,101,186,58,187,68,139),(5,236,102,156,59,188,69,140),(6,237,103,157,60,189,70,141),(7,238,104,158,61,190,71,142),(8,239,105,159,62,191,72,143),(9,240,106,160,32,192,73,144),(10,241,107,161,33,193,74,145),(11,242,108,162,34,194,75,146),(12,243,109,163,35,195,76,147),(13,244,110,164,36,196,77,148),(14,245,111,165,37,197,78,149),(15,246,112,166,38,198,79,150),(16,247,113,167,39,199,80,151),(17,248,114,168,40,200,81,152),(18,218,115,169,41,201,82,153),(19,219,116,170,42,202,83,154),(20,220,117,171,43,203,84,155),(21,221,118,172,44,204,85,125),(22,222,119,173,45,205,86,126),(23,223,120,174,46,206,87,127),(24,224,121,175,47,207,88,128),(25,225,122,176,48,208,89,129),(26,226,123,177,49,209,90,130),(27,227,124,178,50,210,91,131),(28,228,94,179,51,211,92,132),(29,229,95,180,52,212,93,133),(30,230,96,181,53,213,63,134),(31,231,97,182,54,214,64,135)], [(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)], [(1,31),(2,30),(3,29),(4,28),(5,27),(6,26),(7,25),(8,24),(9,23),(10,22),(11,21),(12,20),(13,19),(14,18),(15,17),(32,46),(33,45),(34,44),(35,43),(36,42),(37,41),(38,40),(47,62),(48,61),(49,60),(50,59),(51,58),(52,57),(53,56),(54,55),(63,66),(64,65),(67,93),(68,92),(69,91),(70,90),(71,89),(72,88),(73,87),(74,86),(75,85),(76,84),(77,83),(78,82),(79,81),(94,101),(95,100),(96,99),(97,98),(102,124),(103,123),(104,122),(105,121),(106,120),(107,119),(108,118),(109,117),(110,116),(111,115),(112,114),(125,162),(126,161),(127,160),(128,159),(129,158),(130,157),(131,156),(132,186),(133,185),(134,184),(135,183),(136,182),(137,181),(138,180),(139,179),(140,178),(141,177),(142,176),(143,175),(144,174),(145,173),(146,172),(147,171),(148,170),(149,169),(150,168),(151,167),(152,166),(153,165),(154,164),(155,163),(187,228),(188,227),(189,226),(190,225),(191,224),(192,223),(193,222),(194,221),(195,220),(196,219),(197,218),(198,248),(199,247),(200,246),(201,245),(202,244),(203,243),(204,242),(205,241),(206,240),(207,239),(208,238),(209,237),(210,236),(211,235),(212,234),(213,233),(214,232),(215,231),(216,230),(217,229)]])
130 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 31A | ··· | 31O | 62A | ··· | 62O | 124A | ··· | 124AD | 248A | ··· | 248BH |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 | 248 | ··· | 248 |
| size | 1 | 1 | 62 | 1 | 1 | 62 | 2 | 2 | 62 | 62 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
130 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | M4(2) | D31 | D62 | C4×D31 | C8⋊D31 |
| kernel | C8⋊D31 | C31⋊C8 | C248 | C4×D31 | Dic31 | D62 | C31 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 15 | 15 | 30 | 60 |
Matrix representation of C8⋊D31 ►in GL2(𝔽1489) generated by
| 1480 | 1223 |
| 1367 | 9 |
| 149 | 1 |
| 146 | 1470 |
| 982 | 130 |
| 680 | 507 |
G:=sub<GL(2,GF(1489))| [1480,1367,1223,9],[149,146,1,1470],[982,680,130,507] >;
C8⋊D31 in GAP, Magma, Sage, TeX
C_8\rtimes D_{31} % in TeX
G:=Group("C8:D31"); // GroupNames label
G:=SmallGroup(496,4);
// by ID
G=gap.SmallGroup(496,4);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-31,101,26,42,12004]);
// Polycyclic
G:=Group<a,b,c|a^8=b^31=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations
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