direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D124, C4⋊2D62, C62⋊1D4, C124⋊2C22, D62⋊1C22, C62.3C23, C22.10D62, C31⋊1(C2×D4), (C2×C4)⋊2D31, (C2×C124)⋊3C2, (C22×D31)⋊1C2, C2.4(C22×D31), (C2×C62).10C22, SmallGroup(496,29)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D124
G = < a,b,c | a2=b124=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 880 in 54 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C2, C2, C4, C22, C22, C2×C4, D4, C23, C2×D4, C31, D31, C62, C62, C124, D62, D62, C2×C62, D124, C2×C124, C22×D31, C2×D124
Quotients: C1, C2, C22, D4, C23, C2×D4, D31, D62, D124, C22×D31, C2×D124
►On 248 points
Generators in S248
(1 221)(2 222)(3 223)(4 224)(5 225)(6 226)(7 227)(8 228)(9 229)(10 230)(11 231)(12 232)(13 233)(14 234)(15 235)(16 236)(17 237)(18 238)(19 239)(20 240)(21 241)(22 242)(23 243)(24 244)(25 245)(26 246)(27 247)(28 248)(29 125)(30 126)(31 127)(32 128)(33 129)(34 130)(35 131)(36 132)(37 133)(38 134)(39 135)(40 136)(41 137)(42 138)(43 139)(44 140)(45 141)(46 142)(47 143)(48 144)(49 145)(50 146)(51 147)(52 148)(53 149)(54 150)(55 151)(56 152)(57 153)(58 154)(59 155)(60 156)(61 157)(62 158)(63 159)(64 160)(65 161)(66 162)(67 163)(68 164)(69 165)(70 166)(71 167)(72 168)(73 169)(74 170)(75 171)(76 172)(77 173)(78 174)(79 175)(80 176)(81 177)(82 178)(83 179)(84 180)(85 181)(86 182)(87 183)(88 184)(89 185)(90 186)(91 187)(92 188)(93 189)(94 190)(95 191)(96 192)(97 193)(98 194)(99 195)(100 196)(101 197)(102 198)(103 199)(104 200)(105 201)(106 202)(107 203)(108 204)(109 205)(110 206)(111 207)(112 208)(113 209)(114 210)(115 211)(116 212)(117 213)(118 214)(119 215)(120 216)(121 217)(122 218)(123 219)(124 220)
(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)
(1 124)(2 123)(3 122)(4 121)(5 120)(6 119)(7 118)(8 117)(9 116)(10 115)(11 114)(12 113)(13 112)(14 111)(15 110)(16 109)(17 108)(18 107)(19 106)(20 105)(21 104)(22 103)(23 102)(24 101)(25 100)(26 99)(27 98)(28 97)(29 96)(30 95)(31 94)(32 93)(33 92)(34 91)(35 90)(36 89)(37 88)(38 87)(39 86)(40 85)(41 84)(42 83)(43 82)(44 81)(45 80)(46 79)(47 78)(48 77)(49 76)(50 75)(51 74)(52 73)(53 72)(54 71)(55 70)(56 69)(57 68)(58 67)(59 66)(60 65)(61 64)(62 63)(125 192)(126 191)(127 190)(128 189)(129 188)(130 187)(131 186)(132 185)(133 184)(134 183)(135 182)(136 181)(137 180)(138 179)(139 178)(140 177)(141 176)(142 175)(143 174)(144 173)(145 172)(146 171)(147 170)(148 169)(149 168)(150 167)(151 166)(152 165)(153 164)(154 163)(155 162)(156 161)(157 160)(158 159)(193 248)(194 247)(195 246)(196 245)(197 244)(198 243)(199 242)(200 241)(201 240)(202 239)(203 238)(204 237)(205 236)(206 235)(207 234)(208 233)(209 232)(210 231)(211 230)(212 229)(213 228)(214 227)(215 226)(216 225)(217 224)(218 223)(219 222)(220 221)
G:=sub<Sym(248)| (1,221)(2,222)(3,223)(4,224)(5,225)(6,226)(7,227)(8,228)(9,229)(10,230)(11,231)(12,232)(13,233)(14,234)(15,235)(16,236)(17,237)(18,238)(19,239)(20,240)(21,241)(22,242)(23,243)(24,244)(25,245)(26,246)(27,247)(28,248)(29,125)(30,126)(31,127)(32,128)(33,129)(34,130)(35,131)(36,132)(37,133)(38,134)(39,135)(40,136)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(57,153)(58,154)(59,155)(60,156)(61,157)(62,158)(63,159)(64,160)(65,161)(66,162)(67,163)(68,164)(69,165)(70,166)(71,167)(72,168)(73,169)(74,170)(75,171)(76,172)(77,173)(78,174)(79,175)(80,176)(81,177)(82,178)(83,179)(84,180)(85,181)(86,182)(87,183)(88,184)(89,185)(90,186)(91,187)(92,188)(93,189)(94,190)(95,191)(96,192)(97,193)(98,194)(99,195)(100,196)(101,197)(102,198)(103,199)(104,200)(105,201)(106,202)(107,203)(108,204)(109,205)(110,206)(111,207)(112,208)(113,209)(114,210)(115,211)(116,212)(117,213)(118,214)(119,215)(120,216)(121,217)(122,218)(123,219)(124,220), (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), (1,124)(2,123)(3,122)(4,121)(5,120)(6,119)(7,118)(8,117)(9,116)(10,115)(11,114)(12,113)(13,112)(14,111)(15,110)(16,109)(17,108)(18,107)(19,106)(20,105)(21,104)(22,103)(23,102)(24,101)(25,100)(26,99)(27,98)(28,97)(29,96)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,88)(38,87)(39,86)(40,85)(41,84)(42,83)(43,82)(44,81)(45,80)(46,79)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)(125,192)(126,191)(127,190)(128,189)(129,188)(130,187)(131,186)(132,185)(133,184)(134,183)(135,182)(136,181)(137,180)(138,179)(139,178)(140,177)(141,176)(142,175)(143,174)(144,173)(145,172)(146,171)(147,170)(148,169)(149,168)(150,167)(151,166)(152,165)(153,164)(154,163)(155,162)(156,161)(157,160)(158,159)(193,248)(194,247)(195,246)(196,245)(197,244)(198,243)(199,242)(200,241)(201,240)(202,239)(203,238)(204,237)(205,236)(206,235)(207,234)(208,233)(209,232)(210,231)(211,230)(212,229)(213,228)(214,227)(215,226)(216,225)(217,224)(218,223)(219,222)(220,221)>;
G:=Group( (1,221)(2,222)(3,223)(4,224)(5,225)(6,226)(7,227)(8,228)(9,229)(10,230)(11,231)(12,232)(13,233)(14,234)(15,235)(16,236)(17,237)(18,238)(19,239)(20,240)(21,241)(22,242)(23,243)(24,244)(25,245)(26,246)(27,247)(28,248)(29,125)(30,126)(31,127)(32,128)(33,129)(34,130)(35,131)(36,132)(37,133)(38,134)(39,135)(40,136)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(57,153)(58,154)(59,155)(60,156)(61,157)(62,158)(63,159)(64,160)(65,161)(66,162)(67,163)(68,164)(69,165)(70,166)(71,167)(72,168)(73,169)(74,170)(75,171)(76,172)(77,173)(78,174)(79,175)(80,176)(81,177)(82,178)(83,179)(84,180)(85,181)(86,182)(87,183)(88,184)(89,185)(90,186)(91,187)(92,188)(93,189)(94,190)(95,191)(96,192)(97,193)(98,194)(99,195)(100,196)(101,197)(102,198)(103,199)(104,200)(105,201)(106,202)(107,203)(108,204)(109,205)(110,206)(111,207)(112,208)(113,209)(114,210)(115,211)(116,212)(117,213)(118,214)(119,215)(120,216)(121,217)(122,218)(123,219)(124,220), (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), (1,124)(2,123)(3,122)(4,121)(5,120)(6,119)(7,118)(8,117)(9,116)(10,115)(11,114)(12,113)(13,112)(14,111)(15,110)(16,109)(17,108)(18,107)(19,106)(20,105)(21,104)(22,103)(23,102)(24,101)(25,100)(26,99)(27,98)(28,97)(29,96)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,88)(38,87)(39,86)(40,85)(41,84)(42,83)(43,82)(44,81)(45,80)(46,79)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)(125,192)(126,191)(127,190)(128,189)(129,188)(130,187)(131,186)(132,185)(133,184)(134,183)(135,182)(136,181)(137,180)(138,179)(139,178)(140,177)(141,176)(142,175)(143,174)(144,173)(145,172)(146,171)(147,170)(148,169)(149,168)(150,167)(151,166)(152,165)(153,164)(154,163)(155,162)(156,161)(157,160)(158,159)(193,248)(194,247)(195,246)(196,245)(197,244)(198,243)(199,242)(200,241)(201,240)(202,239)(203,238)(204,237)(205,236)(206,235)(207,234)(208,233)(209,232)(210,231)(211,230)(212,229)(213,228)(214,227)(215,226)(216,225)(217,224)(218,223)(219,222)(220,221) );
G=PermutationGroup([[(1,221),(2,222),(3,223),(4,224),(5,225),(6,226),(7,227),(8,228),(9,229),(10,230),(11,231),(12,232),(13,233),(14,234),(15,235),(16,236),(17,237),(18,238),(19,239),(20,240),(21,241),(22,242),(23,243),(24,244),(25,245),(26,246),(27,247),(28,248),(29,125),(30,126),(31,127),(32,128),(33,129),(34,130),(35,131),(36,132),(37,133),(38,134),(39,135),(40,136),(41,137),(42,138),(43,139),(44,140),(45,141),(46,142),(47,143),(48,144),(49,145),(50,146),(51,147),(52,148),(53,149),(54,150),(55,151),(56,152),(57,153),(58,154),(59,155),(60,156),(61,157),(62,158),(63,159),(64,160),(65,161),(66,162),(67,163),(68,164),(69,165),(70,166),(71,167),(72,168),(73,169),(74,170),(75,171),(76,172),(77,173),(78,174),(79,175),(80,176),(81,177),(82,178),(83,179),(84,180),(85,181),(86,182),(87,183),(88,184),(89,185),(90,186),(91,187),(92,188),(93,189),(94,190),(95,191),(96,192),(97,193),(98,194),(99,195),(100,196),(101,197),(102,198),(103,199),(104,200),(105,201),(106,202),(107,203),(108,204),(109,205),(110,206),(111,207),(112,208),(113,209),(114,210),(115,211),(116,212),(117,213),(118,214),(119,215),(120,216),(121,217),(122,218),(123,219),(124,220)], [(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)], [(1,124),(2,123),(3,122),(4,121),(5,120),(6,119),(7,118),(8,117),(9,116),(10,115),(11,114),(12,113),(13,112),(14,111),(15,110),(16,109),(17,108),(18,107),(19,106),(20,105),(21,104),(22,103),(23,102),(24,101),(25,100),(26,99),(27,98),(28,97),(29,96),(30,95),(31,94),(32,93),(33,92),(34,91),(35,90),(36,89),(37,88),(38,87),(39,86),(40,85),(41,84),(42,83),(43,82),(44,81),(45,80),(46,79),(47,78),(48,77),(49,76),(50,75),(51,74),(52,73),(53,72),(54,71),(55,70),(56,69),(57,68),(58,67),(59,66),(60,65),(61,64),(62,63),(125,192),(126,191),(127,190),(128,189),(129,188),(130,187),(131,186),(132,185),(133,184),(134,183),(135,182),(136,181),(137,180),(138,179),(139,178),(140,177),(141,176),(142,175),(143,174),(144,173),(145,172),(146,171),(147,170),(148,169),(149,168),(150,167),(151,166),(152,165),(153,164),(154,163),(155,162),(156,161),(157,160),(158,159),(193,248),(194,247),(195,246),(196,245),(197,244),(198,243),(199,242),(200,241),(201,240),(202,239),(203,238),(204,237),(205,236),(206,235),(207,234),(208,233),(209,232),(210,231),(211,230),(212,229),(213,228),(214,227),(215,226),(216,225),(217,224),(218,223),(219,222),(220,221)]])
130 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 31A | ··· | 31O | 62A | ··· | 62AS | 124A | ··· | 124BH |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 |
| size | 1 | 1 | 1 | 1 | 62 | 62 | 62 | 62 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
130 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | D4 | D31 | D62 | D62 | D124 |
| kernel | C2×D124 | D124 | C2×C124 | C22×D31 | C62 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 2 | 15 | 30 | 15 | 60 |
Matrix representation of C2×D124 ►in GL3(𝔽373) generated by
| 372 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 113 | 30 |
| 0 | 343 | 200 |
| 372 | 0 | 0 |
| 0 | 113 | 30 |
| 0 | 22 | 260 |
G:=sub<GL(3,GF(373))| [372,0,0,0,1,0,0,0,1],[1,0,0,0,113,343,0,30,200],[372,0,0,0,113,22,0,30,260] >;
C2×D124 in GAP, Magma, Sage, TeX
C_2\times D_{124} % in TeX
G:=Group("C2xD124"); // GroupNames label
G:=SmallGroup(496,29);
// by ID
G=gap.SmallGroup(496,29);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-31,182,42,12004]);
// Polycyclic
G:=Group<a,b,c|a^2=b^124=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations