metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D62⋊C4, C62.6D4, C2.2D124, C22.6D62, (C2×C4)⋊1D31, (C2×C124)⋊1C2, C62.5(C2×C4), C2.5(C4×D31), C31⋊1(C22⋊C4), (C22×D31).C2, (C2×Dic31)⋊1C2, C2.2(C31⋊D4), (C2×C62).6C22, SmallGroup(496,13)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D62⋊C4
G = < a,b,c | a62=b2=c4=1, bab=a-1, ac=ca, cbc-1=a31b >
Smallest permutation representation of D62⋊C4
►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 62)(2 61)(3 60)(4 59)(5 58)(6 57)(7 56)(8 55)(9 54)(10 53)(11 52)(12 51)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(31 32)(63 100)(64 99)(65 98)(66 97)(67 96)(68 95)(69 94)(70 93)(71 92)(72 91)(73 90)(74 89)(75 88)(76 87)(77 86)(78 85)(79 84)(80 83)(81 82)(101 124)(102 123)(103 122)(104 121)(105 120)(106 119)(107 118)(108 117)(109 116)(110 115)(111 114)(112 113)(125 151)(126 150)(127 149)(128 148)(129 147)(130 146)(131 145)(132 144)(133 143)(134 142)(135 141)(136 140)(137 139)(152 186)(153 185)(154 184)(155 183)(156 182)(157 181)(158 180)(159 179)(160 178)(161 177)(162 176)(163 175)(164 174)(165 173)(166 172)(167 171)(168 170)(187 225)(188 224)(189 223)(190 222)(191 221)(192 220)(193 219)(194 218)(195 217)(196 216)(197 215)(198 214)(199 213)(200 212)(201 211)(202 210)(203 209)(204 208)(205 207)(226 248)(227 247)(228 246)(229 245)(230 244)(231 243)(232 242)(233 241)(234 240)(235 239)(236 238)
(1 191 82 185)(2 192 83 186)(3 193 84 125)(4 194 85 126)(5 195 86 127)(6 196 87 128)(7 197 88 129)(8 198 89 130)(9 199 90 131)(10 200 91 132)(11 201 92 133)(12 202 93 134)(13 203 94 135)(14 204 95 136)(15 205 96 137)(16 206 97 138)(17 207 98 139)(18 208 99 140)(19 209 100 141)(20 210 101 142)(21 211 102 143)(22 212 103 144)(23 213 104 145)(24 214 105 146)(25 215 106 147)(26 216 107 148)(27 217 108 149)(28 218 109 150)(29 219 110 151)(30 220 111 152)(31 221 112 153)(32 222 113 154)(33 223 114 155)(34 224 115 156)(35 225 116 157)(36 226 117 158)(37 227 118 159)(38 228 119 160)(39 229 120 161)(40 230 121 162)(41 231 122 163)(42 232 123 164)(43 233 124 165)(44 234 63 166)(45 235 64 167)(46 236 65 168)(47 237 66 169)(48 238 67 170)(49 239 68 171)(50 240 69 172)(51 241 70 173)(52 242 71 174)(53 243 72 175)(54 244 73 176)(55 245 74 177)(56 246 75 178)(57 247 76 179)(58 248 77 180)(59 187 78 181)(60 188 79 182)(61 189 80 183)(62 190 81 184)
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), (1,62)(2,61)(3,60)(4,59)(5,58)(6,57)(7,56)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)(101,124)(102,123)(103,122)(104,121)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113)(125,151)(126,150)(127,149)(128,148)(129,147)(130,146)(131,145)(132,144)(133,143)(134,142)(135,141)(136,140)(137,139)(152,186)(153,185)(154,184)(155,183)(156,182)(157,181)(158,180)(159,179)(160,178)(161,177)(162,176)(163,175)(164,174)(165,173)(166,172)(167,171)(168,170)(187,225)(188,224)(189,223)(190,222)(191,221)(192,220)(193,219)(194,218)(195,217)(196,216)(197,215)(198,214)(199,213)(200,212)(201,211)(202,210)(203,209)(204,208)(205,207)(226,248)(227,247)(228,246)(229,245)(230,244)(231,243)(232,242)(233,241)(234,240)(235,239)(236,238), (1,191,82,185)(2,192,83,186)(3,193,84,125)(4,194,85,126)(5,195,86,127)(6,196,87,128)(7,197,88,129)(8,198,89,130)(9,199,90,131)(10,200,91,132)(11,201,92,133)(12,202,93,134)(13,203,94,135)(14,204,95,136)(15,205,96,137)(16,206,97,138)(17,207,98,139)(18,208,99,140)(19,209,100,141)(20,210,101,142)(21,211,102,143)(22,212,103,144)(23,213,104,145)(24,214,105,146)(25,215,106,147)(26,216,107,148)(27,217,108,149)(28,218,109,150)(29,219,110,151)(30,220,111,152)(31,221,112,153)(32,222,113,154)(33,223,114,155)(34,224,115,156)(35,225,116,157)(36,226,117,158)(37,227,118,159)(38,228,119,160)(39,229,120,161)(40,230,121,162)(41,231,122,163)(42,232,123,164)(43,233,124,165)(44,234,63,166)(45,235,64,167)(46,236,65,168)(47,237,66,169)(48,238,67,170)(49,239,68,171)(50,240,69,172)(51,241,70,173)(52,242,71,174)(53,243,72,175)(54,244,73,176)(55,245,74,177)(56,246,75,178)(57,247,76,179)(58,248,77,180)(59,187,78,181)(60,188,79,182)(61,189,80,183)(62,190,81,184)>;
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), (1,62)(2,61)(3,60)(4,59)(5,58)(6,57)(7,56)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)(101,124)(102,123)(103,122)(104,121)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113)(125,151)(126,150)(127,149)(128,148)(129,147)(130,146)(131,145)(132,144)(133,143)(134,142)(135,141)(136,140)(137,139)(152,186)(153,185)(154,184)(155,183)(156,182)(157,181)(158,180)(159,179)(160,178)(161,177)(162,176)(163,175)(164,174)(165,173)(166,172)(167,171)(168,170)(187,225)(188,224)(189,223)(190,222)(191,221)(192,220)(193,219)(194,218)(195,217)(196,216)(197,215)(198,214)(199,213)(200,212)(201,211)(202,210)(203,209)(204,208)(205,207)(226,248)(227,247)(228,246)(229,245)(230,244)(231,243)(232,242)(233,241)(234,240)(235,239)(236,238), (1,191,82,185)(2,192,83,186)(3,193,84,125)(4,194,85,126)(5,195,86,127)(6,196,87,128)(7,197,88,129)(8,198,89,130)(9,199,90,131)(10,200,91,132)(11,201,92,133)(12,202,93,134)(13,203,94,135)(14,204,95,136)(15,205,96,137)(16,206,97,138)(17,207,98,139)(18,208,99,140)(19,209,100,141)(20,210,101,142)(21,211,102,143)(22,212,103,144)(23,213,104,145)(24,214,105,146)(25,215,106,147)(26,216,107,148)(27,217,108,149)(28,218,109,150)(29,219,110,151)(30,220,111,152)(31,221,112,153)(32,222,113,154)(33,223,114,155)(34,224,115,156)(35,225,116,157)(36,226,117,158)(37,227,118,159)(38,228,119,160)(39,229,120,161)(40,230,121,162)(41,231,122,163)(42,232,123,164)(43,233,124,165)(44,234,63,166)(45,235,64,167)(46,236,65,168)(47,237,66,169)(48,238,67,170)(49,239,68,171)(50,240,69,172)(51,241,70,173)(52,242,71,174)(53,243,72,175)(54,244,73,176)(55,245,74,177)(56,246,75,178)(57,247,76,179)(58,248,77,180)(59,187,78,181)(60,188,79,182)(61,189,80,183)(62,190,81,184) );
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)], [(1,62),(2,61),(3,60),(4,59),(5,58),(6,57),(7,56),(8,55),(9,54),(10,53),(11,52),(12,51),(13,50),(14,49),(15,48),(16,47),(17,46),(18,45),(19,44),(20,43),(21,42),(22,41),(23,40),(24,39),(25,38),(26,37),(27,36),(28,35),(29,34),(30,33),(31,32),(63,100),(64,99),(65,98),(66,97),(67,96),(68,95),(69,94),(70,93),(71,92),(72,91),(73,90),(74,89),(75,88),(76,87),(77,86),(78,85),(79,84),(80,83),(81,82),(101,124),(102,123),(103,122),(104,121),(105,120),(106,119),(107,118),(108,117),(109,116),(110,115),(111,114),(112,113),(125,151),(126,150),(127,149),(128,148),(129,147),(130,146),(131,145),(132,144),(133,143),(134,142),(135,141),(136,140),(137,139),(152,186),(153,185),(154,184),(155,183),(156,182),(157,181),(158,180),(159,179),(160,178),(161,177),(162,176),(163,175),(164,174),(165,173),(166,172),(167,171),(168,170),(187,225),(188,224),(189,223),(190,222),(191,221),(192,220),(193,219),(194,218),(195,217),(196,216),(197,215),(198,214),(199,213),(200,212),(201,211),(202,210),(203,209),(204,208),(205,207),(226,248),(227,247),(228,246),(229,245),(230,244),(231,243),(232,242),(233,241),(234,240),(235,239),(236,238)], [(1,191,82,185),(2,192,83,186),(3,193,84,125),(4,194,85,126),(5,195,86,127),(6,196,87,128),(7,197,88,129),(8,198,89,130),(9,199,90,131),(10,200,91,132),(11,201,92,133),(12,202,93,134),(13,203,94,135),(14,204,95,136),(15,205,96,137),(16,206,97,138),(17,207,98,139),(18,208,99,140),(19,209,100,141),(20,210,101,142),(21,211,102,143),(22,212,103,144),(23,213,104,145),(24,214,105,146),(25,215,106,147),(26,216,107,148),(27,217,108,149),(28,218,109,150),(29,219,110,151),(30,220,111,152),(31,221,112,153),(32,222,113,154),(33,223,114,155),(34,224,115,156),(35,225,116,157),(36,226,117,158),(37,227,118,159),(38,228,119,160),(39,229,120,161),(40,230,121,162),(41,231,122,163),(42,232,123,164),(43,233,124,165),(44,234,63,166),(45,235,64,167),(46,236,65,168),(47,237,66,169),(48,238,67,170),(49,239,68,171),(50,240,69,172),(51,241,70,173),(52,242,71,174),(53,243,72,175),(54,244,73,176),(55,245,74,177),(56,246,75,178),(57,247,76,179),(58,248,77,180),(59,187,78,181),(60,188,79,182),(61,189,80,183),(62,190,81,184)]])
130 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 31A | ··· | 31O | 62A | ··· | 62AS | 124A | ··· | 124BH |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 |
| size | 1 | 1 | 1 | 1 | 62 | 62 | 2 | 2 | 62 | 62 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
130 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D31 | D62 | C4×D31 | D124 | C31⋊D4 |
| kernel | D62⋊C4 | C2×Dic31 | C2×C124 | C22×D31 | D62 | C62 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 15 | 15 | 30 | 30 | 30 |
Matrix representation of D62⋊C4 ►in GL3(𝔽373) generated by
| 1 | 0 | 0 |
| 0 | 154 | 154 |
| 0 | 219 | 328 |
| 372 | 0 | 0 |
| 0 | 154 | 154 |
| 0 | 328 | 219 |
| 104 | 0 | 0 |
| 0 | 296 | 102 |
| 0 | 271 | 77 |
G:=sub<GL(3,GF(373))| [1,0,0,0,154,219,0,154,328],[372,0,0,0,154,328,0,154,219],[104,0,0,0,296,271,0,102,77] >;
D62⋊C4 in GAP, Magma, Sage, TeX
D_{62}\rtimes C_4 % in TeX
G:=Group("D62:C4"); // GroupNames label
G:=SmallGroup(496,13);
// by ID
G=gap.SmallGroup(496,13);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-31,101,26,12004]);
// Polycyclic
G:=Group<a,b,c|a^62=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^31*b>;
// generators/relations
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