direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C62, C23⋊C62, C124⋊4C22, C62.11C23, C4⋊(C2×C62), (C2×C4)⋊2C62, C22⋊(C2×C62), (C2×C124)⋊6C2, (C22×C62)⋊1C2, (C2×C62)⋊2C22, C2.1(C22×C62), SmallGroup(496,38)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C62
G = < a,b,c | a62=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 70 in 54 conjugacy classes, 38 normal (10 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, D4, C23, C2×D4, C31, C62, C62, C62, C124, C2×C62, C2×C62, C2×C62, C2×C124, D4×C31, C22×C62, D4×C62
Quotients: C1, C2, C22, D4, C23, C2×D4, C31, C62, C2×C62, D4×C31, C22×C62, D4×C62
►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)
(1 248 129 120)(2 187 130 121)(3 188 131 122)(4 189 132 123)(5 190 133 124)(6 191 134 63)(7 192 135 64)(8 193 136 65)(9 194 137 66)(10 195 138 67)(11 196 139 68)(12 197 140 69)(13 198 141 70)(14 199 142 71)(15 200 143 72)(16 201 144 73)(17 202 145 74)(18 203 146 75)(19 204 147 76)(20 205 148 77)(21 206 149 78)(22 207 150 79)(23 208 151 80)(24 209 152 81)(25 210 153 82)(26 211 154 83)(27 212 155 84)(28 213 156 85)(29 214 157 86)(30 215 158 87)(31 216 159 88)(32 217 160 89)(33 218 161 90)(34 219 162 91)(35 220 163 92)(36 221 164 93)(37 222 165 94)(38 223 166 95)(39 224 167 96)(40 225 168 97)(41 226 169 98)(42 227 170 99)(43 228 171 100)(44 229 172 101)(45 230 173 102)(46 231 174 103)(47 232 175 104)(48 233 176 105)(49 234 177 106)(50 235 178 107)(51 236 179 108)(52 237 180 109)(53 238 181 110)(54 239 182 111)(55 240 183 112)(56 241 184 113)(57 242 185 114)(58 243 186 115)(59 244 125 116)(60 245 126 117)(61 246 127 118)(62 247 128 119)
(1 160)(2 161)(3 162)(4 163)(5 164)(6 165)(7 166)(8 167)(9 168)(10 169)(11 170)(12 171)(13 172)(14 173)(15 174)(16 175)(17 176)(18 177)(19 178)(20 179)(21 180)(22 181)(23 182)(24 183)(25 184)(26 185)(27 186)(28 125)(29 126)(30 127)(31 128)(32 129)(33 130)(34 131)(35 132)(36 133)(37 134)(38 135)(39 136)(40 137)(41 138)(42 139)(43 140)(44 141)(45 142)(46 143)(47 144)(48 145)(49 146)(50 147)(51 148)(52 149)(53 150)(54 151)(55 152)(56 153)(57 154)(58 155)(59 156)(60 157)(61 158)(62 159)(63 94)(64 95)(65 96)(66 97)(67 98)(68 99)(69 100)(70 101)(71 102)(72 103)(73 104)(74 105)(75 106)(76 107)(77 108)(78 109)(79 110)(80 111)(81 112)(82 113)(83 114)(84 115)(85 116)(86 117)(87 118)(88 119)(89 120)(90 121)(91 122)(92 123)(93 124)(187 218)(188 219)(189 220)(190 221)(191 222)(192 223)(193 224)(194 225)(195 226)(196 227)(197 228)(198 229)(199 230)(200 231)(201 232)(202 233)(203 234)(204 235)(205 236)(206 237)(207 238)(208 239)(209 240)(210 241)(211 242)(212 243)(213 244)(214 245)(215 246)(216 247)(217 248)
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), (1,248,129,120)(2,187,130,121)(3,188,131,122)(4,189,132,123)(5,190,133,124)(6,191,134,63)(7,192,135,64)(8,193,136,65)(9,194,137,66)(10,195,138,67)(11,196,139,68)(12,197,140,69)(13,198,141,70)(14,199,142,71)(15,200,143,72)(16,201,144,73)(17,202,145,74)(18,203,146,75)(19,204,147,76)(20,205,148,77)(21,206,149,78)(22,207,150,79)(23,208,151,80)(24,209,152,81)(25,210,153,82)(26,211,154,83)(27,212,155,84)(28,213,156,85)(29,214,157,86)(30,215,158,87)(31,216,159,88)(32,217,160,89)(33,218,161,90)(34,219,162,91)(35,220,163,92)(36,221,164,93)(37,222,165,94)(38,223,166,95)(39,224,167,96)(40,225,168,97)(41,226,169,98)(42,227,170,99)(43,228,171,100)(44,229,172,101)(45,230,173,102)(46,231,174,103)(47,232,175,104)(48,233,176,105)(49,234,177,106)(50,235,178,107)(51,236,179,108)(52,237,180,109)(53,238,181,110)(54,239,182,111)(55,240,183,112)(56,241,184,113)(57,242,185,114)(58,243,186,115)(59,244,125,116)(60,245,126,117)(61,246,127,118)(62,247,128,119), (1,160)(2,161)(3,162)(4,163)(5,164)(6,165)(7,166)(8,167)(9,168)(10,169)(11,170)(12,171)(13,172)(14,173)(15,174)(16,175)(17,176)(18,177)(19,178)(20,179)(21,180)(22,181)(23,182)(24,183)(25,184)(26,185)(27,186)(28,125)(29,126)(30,127)(31,128)(32,129)(33,130)(34,131)(35,132)(36,133)(37,134)(38,135)(39,136)(40,137)(41,138)(42,139)(43,140)(44,141)(45,142)(46,143)(47,144)(48,145)(49,146)(50,147)(51,148)(52,149)(53,150)(54,151)(55,152)(56,153)(57,154)(58,155)(59,156)(60,157)(61,158)(62,159)(63,94)(64,95)(65,96)(66,97)(67,98)(68,99)(69,100)(70,101)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111)(81,112)(82,113)(83,114)(84,115)(85,116)(86,117)(87,118)(88,119)(89,120)(90,121)(91,122)(92,123)(93,124)(187,218)(188,219)(189,220)(190,221)(191,222)(192,223)(193,224)(194,225)(195,226)(196,227)(197,228)(198,229)(199,230)(200,231)(201,232)(202,233)(203,234)(204,235)(205,236)(206,237)(207,238)(208,239)(209,240)(210,241)(211,242)(212,243)(213,244)(214,245)(215,246)(216,247)(217,248)>;
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), (1,248,129,120)(2,187,130,121)(3,188,131,122)(4,189,132,123)(5,190,133,124)(6,191,134,63)(7,192,135,64)(8,193,136,65)(9,194,137,66)(10,195,138,67)(11,196,139,68)(12,197,140,69)(13,198,141,70)(14,199,142,71)(15,200,143,72)(16,201,144,73)(17,202,145,74)(18,203,146,75)(19,204,147,76)(20,205,148,77)(21,206,149,78)(22,207,150,79)(23,208,151,80)(24,209,152,81)(25,210,153,82)(26,211,154,83)(27,212,155,84)(28,213,156,85)(29,214,157,86)(30,215,158,87)(31,216,159,88)(32,217,160,89)(33,218,161,90)(34,219,162,91)(35,220,163,92)(36,221,164,93)(37,222,165,94)(38,223,166,95)(39,224,167,96)(40,225,168,97)(41,226,169,98)(42,227,170,99)(43,228,171,100)(44,229,172,101)(45,230,173,102)(46,231,174,103)(47,232,175,104)(48,233,176,105)(49,234,177,106)(50,235,178,107)(51,236,179,108)(52,237,180,109)(53,238,181,110)(54,239,182,111)(55,240,183,112)(56,241,184,113)(57,242,185,114)(58,243,186,115)(59,244,125,116)(60,245,126,117)(61,246,127,118)(62,247,128,119), (1,160)(2,161)(3,162)(4,163)(5,164)(6,165)(7,166)(8,167)(9,168)(10,169)(11,170)(12,171)(13,172)(14,173)(15,174)(16,175)(17,176)(18,177)(19,178)(20,179)(21,180)(22,181)(23,182)(24,183)(25,184)(26,185)(27,186)(28,125)(29,126)(30,127)(31,128)(32,129)(33,130)(34,131)(35,132)(36,133)(37,134)(38,135)(39,136)(40,137)(41,138)(42,139)(43,140)(44,141)(45,142)(46,143)(47,144)(48,145)(49,146)(50,147)(51,148)(52,149)(53,150)(54,151)(55,152)(56,153)(57,154)(58,155)(59,156)(60,157)(61,158)(62,159)(63,94)(64,95)(65,96)(66,97)(67,98)(68,99)(69,100)(70,101)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111)(81,112)(82,113)(83,114)(84,115)(85,116)(86,117)(87,118)(88,119)(89,120)(90,121)(91,122)(92,123)(93,124)(187,218)(188,219)(189,220)(190,221)(191,222)(192,223)(193,224)(194,225)(195,226)(196,227)(197,228)(198,229)(199,230)(200,231)(201,232)(202,233)(203,234)(204,235)(205,236)(206,237)(207,238)(208,239)(209,240)(210,241)(211,242)(212,243)(213,244)(214,245)(215,246)(216,247)(217,248) );
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)], [(1,248,129,120),(2,187,130,121),(3,188,131,122),(4,189,132,123),(5,190,133,124),(6,191,134,63),(7,192,135,64),(8,193,136,65),(9,194,137,66),(10,195,138,67),(11,196,139,68),(12,197,140,69),(13,198,141,70),(14,199,142,71),(15,200,143,72),(16,201,144,73),(17,202,145,74),(18,203,146,75),(19,204,147,76),(20,205,148,77),(21,206,149,78),(22,207,150,79),(23,208,151,80),(24,209,152,81),(25,210,153,82),(26,211,154,83),(27,212,155,84),(28,213,156,85),(29,214,157,86),(30,215,158,87),(31,216,159,88),(32,217,160,89),(33,218,161,90),(34,219,162,91),(35,220,163,92),(36,221,164,93),(37,222,165,94),(38,223,166,95),(39,224,167,96),(40,225,168,97),(41,226,169,98),(42,227,170,99),(43,228,171,100),(44,229,172,101),(45,230,173,102),(46,231,174,103),(47,232,175,104),(48,233,176,105),(49,234,177,106),(50,235,178,107),(51,236,179,108),(52,237,180,109),(53,238,181,110),(54,239,182,111),(55,240,183,112),(56,241,184,113),(57,242,185,114),(58,243,186,115),(59,244,125,116),(60,245,126,117),(61,246,127,118),(62,247,128,119)], [(1,160),(2,161),(3,162),(4,163),(5,164),(6,165),(7,166),(8,167),(9,168),(10,169),(11,170),(12,171),(13,172),(14,173),(15,174),(16,175),(17,176),(18,177),(19,178),(20,179),(21,180),(22,181),(23,182),(24,183),(25,184),(26,185),(27,186),(28,125),(29,126),(30,127),(31,128),(32,129),(33,130),(34,131),(35,132),(36,133),(37,134),(38,135),(39,136),(40,137),(41,138),(42,139),(43,140),(44,141),(45,142),(46,143),(47,144),(48,145),(49,146),(50,147),(51,148),(52,149),(53,150),(54,151),(55,152),(56,153),(57,154),(58,155),(59,156),(60,157),(61,158),(62,159),(63,94),(64,95),(65,96),(66,97),(67,98),(68,99),(69,100),(70,101),(71,102),(72,103),(73,104),(74,105),(75,106),(76,107),(77,108),(78,109),(79,110),(80,111),(81,112),(82,113),(83,114),(84,115),(85,116),(86,117),(87,118),(88,119),(89,120),(90,121),(91,122),(92,123),(93,124),(187,218),(188,219),(189,220),(190,221),(191,222),(192,223),(193,224),(194,225),(195,226),(196,227),(197,228),(198,229),(199,230),(200,231),(201,232),(202,233),(203,234),(204,235),(205,236),(206,237),(207,238),(208,239),(209,240),(210,241),(211,242),(212,243),(213,244),(214,245),(215,246),(216,247),(217,248)]])
310 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 31A | ··· | 31AD | 62A | ··· | 62CL | 62CM | ··· | 62HB | 124A | ··· | 124BH |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 31 | ··· | 31 | 62 | ··· | 62 | 62 | ··· | 62 | 124 | ··· | 124 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
310 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C31 | C62 | C62 | C62 | D4 | D4×C31 |
| kernel | D4×C62 | C2×C124 | D4×C31 | C22×C62 | C2×D4 | C2×C4 | D4 | C23 | C62 | C2 |
| # reps | 1 | 1 | 4 | 2 | 30 | 30 | 120 | 60 | 2 | 60 |
Matrix representation of D4×C62 ►in GL3(𝔽373) generated by
| 372 | 0 | 0 |
| 0 | 356 | 0 |
| 0 | 0 | 356 |
| 372 | 0 | 0 |
| 0 | 372 | 2 |
| 0 | 372 | 1 |
| 1 | 0 | 0 |
| 0 | 372 | 0 |
| 0 | 372 | 1 |
G:=sub<GL(3,GF(373))| [372,0,0,0,356,0,0,0,356],[372,0,0,0,372,372,0,2,1],[1,0,0,0,372,372,0,0,1] >;
D4×C62 in GAP, Magma, Sage, TeX
D_4\times C_{62} % in TeX
G:=Group("D4xC62"); // GroupNames label
G:=SmallGroup(496,38);
// by ID
G=gap.SmallGroup(496,38);
# by ID
G:=PCGroup([5,-2,-2,-2,-31,-2,2501]);
// Polycyclic
G:=Group<a,b,c|a^62=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations