direct product, abelian, monomial, 2-elementary
Aliases: C2×C150, SmallGroup(300,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C150 |
| C1 — C2×C150 |
| C1 — C2×C150 |
Generators and relations for C2×C150
G = < a,b | a2=b150=1, ab=ba >
Smallest permutation representation of C2×C150
►Regular action on 300 points
Generators in S300
(1 152)(2 153)(3 154)(4 155)(5 156)(6 157)(7 158)(8 159)(9 160)(10 161)(11 162)(12 163)(13 164)(14 165)(15 166)(16 167)(17 168)(18 169)(19 170)(20 171)(21 172)(22 173)(23 174)(24 175)(25 176)(26 177)(27 178)(28 179)(29 180)(30 181)(31 182)(32 183)(33 184)(34 185)(35 186)(36 187)(37 188)(38 189)(39 190)(40 191)(41 192)(42 193)(43 194)(44 195)(45 196)(46 197)(47 198)(48 199)(49 200)(50 201)(51 202)(52 203)(53 204)(54 205)(55 206)(56 207)(57 208)(58 209)(59 210)(60 211)(61 212)(62 213)(63 214)(64 215)(65 216)(66 217)(67 218)(68 219)(69 220)(70 221)(71 222)(72 223)(73 224)(74 225)(75 226)(76 227)(77 228)(78 229)(79 230)(80 231)(81 232)(82 233)(83 234)(84 235)(85 236)(86 237)(87 238)(88 239)(89 240)(90 241)(91 242)(92 243)(93 244)(94 245)(95 246)(96 247)(97 248)(98 249)(99 250)(100 251)(101 252)(102 253)(103 254)(104 255)(105 256)(106 257)(107 258)(108 259)(109 260)(110 261)(111 262)(112 263)(113 264)(114 265)(115 266)(116 267)(117 268)(118 269)(119 270)(120 271)(121 272)(122 273)(123 274)(124 275)(125 276)(126 277)(127 278)(128 279)(129 280)(130 281)(131 282)(132 283)(133 284)(134 285)(135 286)(136 287)(137 288)(138 289)(139 290)(140 291)(141 292)(142 293)(143 294)(144 295)(145 296)(146 297)(147 298)(148 299)(149 300)(150 151)
(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)(151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300)
G:=sub<Sym(300)| (1,152)(2,153)(3,154)(4,155)(5,156)(6,157)(7,158)(8,159)(9,160)(10,161)(11,162)(12,163)(13,164)(14,165)(15,166)(16,167)(17,168)(18,169)(19,170)(20,171)(21,172)(22,173)(23,174)(24,175)(25,176)(26,177)(27,178)(28,179)(29,180)(30,181)(31,182)(32,183)(33,184)(34,185)(35,186)(36,187)(37,188)(38,189)(39,190)(40,191)(41,192)(42,193)(43,194)(44,195)(45,196)(46,197)(47,198)(48,199)(49,200)(50,201)(51,202)(52,203)(53,204)(54,205)(55,206)(56,207)(57,208)(58,209)(59,210)(60,211)(61,212)(62,213)(63,214)(64,215)(65,216)(66,217)(67,218)(68,219)(69,220)(70,221)(71,222)(72,223)(73,224)(74,225)(75,226)(76,227)(77,228)(78,229)(79,230)(80,231)(81,232)(82,233)(83,234)(84,235)(85,236)(86,237)(87,238)(88,239)(89,240)(90,241)(91,242)(92,243)(93,244)(94,245)(95,246)(96,247)(97,248)(98,249)(99,250)(100,251)(101,252)(102,253)(103,254)(104,255)(105,256)(106,257)(107,258)(108,259)(109,260)(110,261)(111,262)(112,263)(113,264)(114,265)(115,266)(116,267)(117,268)(118,269)(119,270)(120,271)(121,272)(122,273)(123,274)(124,275)(125,276)(126,277)(127,278)(128,279)(129,280)(130,281)(131,282)(132,283)(133,284)(134,285)(135,286)(136,287)(137,288)(138,289)(139,290)(140,291)(141,292)(142,293)(143,294)(144,295)(145,296)(146,297)(147,298)(148,299)(149,300)(150,151), (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)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300)>;
G:=Group( (1,152)(2,153)(3,154)(4,155)(5,156)(6,157)(7,158)(8,159)(9,160)(10,161)(11,162)(12,163)(13,164)(14,165)(15,166)(16,167)(17,168)(18,169)(19,170)(20,171)(21,172)(22,173)(23,174)(24,175)(25,176)(26,177)(27,178)(28,179)(29,180)(30,181)(31,182)(32,183)(33,184)(34,185)(35,186)(36,187)(37,188)(38,189)(39,190)(40,191)(41,192)(42,193)(43,194)(44,195)(45,196)(46,197)(47,198)(48,199)(49,200)(50,201)(51,202)(52,203)(53,204)(54,205)(55,206)(56,207)(57,208)(58,209)(59,210)(60,211)(61,212)(62,213)(63,214)(64,215)(65,216)(66,217)(67,218)(68,219)(69,220)(70,221)(71,222)(72,223)(73,224)(74,225)(75,226)(76,227)(77,228)(78,229)(79,230)(80,231)(81,232)(82,233)(83,234)(84,235)(85,236)(86,237)(87,238)(88,239)(89,240)(90,241)(91,242)(92,243)(93,244)(94,245)(95,246)(96,247)(97,248)(98,249)(99,250)(100,251)(101,252)(102,253)(103,254)(104,255)(105,256)(106,257)(107,258)(108,259)(109,260)(110,261)(111,262)(112,263)(113,264)(114,265)(115,266)(116,267)(117,268)(118,269)(119,270)(120,271)(121,272)(122,273)(123,274)(124,275)(125,276)(126,277)(127,278)(128,279)(129,280)(130,281)(131,282)(132,283)(133,284)(134,285)(135,286)(136,287)(137,288)(138,289)(139,290)(140,291)(141,292)(142,293)(143,294)(144,295)(145,296)(146,297)(147,298)(148,299)(149,300)(150,151), (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)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300) );
G=PermutationGroup([[(1,152),(2,153),(3,154),(4,155),(5,156),(6,157),(7,158),(8,159),(9,160),(10,161),(11,162),(12,163),(13,164),(14,165),(15,166),(16,167),(17,168),(18,169),(19,170),(20,171),(21,172),(22,173),(23,174),(24,175),(25,176),(26,177),(27,178),(28,179),(29,180),(30,181),(31,182),(32,183),(33,184),(34,185),(35,186),(36,187),(37,188),(38,189),(39,190),(40,191),(41,192),(42,193),(43,194),(44,195),(45,196),(46,197),(47,198),(48,199),(49,200),(50,201),(51,202),(52,203),(53,204),(54,205),(55,206),(56,207),(57,208),(58,209),(59,210),(60,211),(61,212),(62,213),(63,214),(64,215),(65,216),(66,217),(67,218),(68,219),(69,220),(70,221),(71,222),(72,223),(73,224),(74,225),(75,226),(76,227),(77,228),(78,229),(79,230),(80,231),(81,232),(82,233),(83,234),(84,235),(85,236),(86,237),(87,238),(88,239),(89,240),(90,241),(91,242),(92,243),(93,244),(94,245),(95,246),(96,247),(97,248),(98,249),(99,250),(100,251),(101,252),(102,253),(103,254),(104,255),(105,256),(106,257),(107,258),(108,259),(109,260),(110,261),(111,262),(112,263),(113,264),(114,265),(115,266),(116,267),(117,268),(118,269),(119,270),(120,271),(121,272),(122,273),(123,274),(124,275),(125,276),(126,277),(127,278),(128,279),(129,280),(130,281),(131,282),(132,283),(133,284),(134,285),(135,286),(136,287),(137,288),(138,289),(139,290),(140,291),(141,292),(142,293),(143,294),(144,295),(145,296),(146,297),(147,298),(148,299),(149,300),(150,151)], [(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),(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300)]])
300 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 10A | ··· | 10L | 15A | ··· | 15H | 25A | ··· | 25T | 30A | ··· | 30X | 50A | ··· | 50BH | 75A | ··· | 75AN | 150A | ··· | 150DP |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 15 | ··· | 15 | 25 | ··· | 25 | 30 | ··· | 30 | 50 | ··· | 50 | 75 | ··· | 75 | 150 | ··· | 150 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
300 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C25 | C30 | C50 | C75 | C150 |
| kernel | C2×C150 | C150 | C2×C50 | C2×C30 | C50 | C30 | C2×C10 | C2×C6 | C10 | C6 | C22 | C2 |
| # reps | 1 | 3 | 2 | 4 | 6 | 12 | 8 | 20 | 24 | 60 | 40 | 120 |
Matrix representation of C2×C150 ►in GL2(𝔽151) generated by
| 150 | 0 |
| 0 | 150 |
| 33 | 0 |
| 0 | 17 |
G:=sub<GL(2,GF(151))| [150,0,0,150],[33,0,0,17] >;
C2×C150 in GAP, Magma, Sage, TeX
C_2\times C_{150} % in TeX
G:=Group("C2xC150"); // GroupNames label
G:=SmallGroup(300,12);
// by ID
G=gap.SmallGroup(300,12);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-5,118]);
// Polycyclic
G:=Group<a,b|a^2=b^150=1,a*b=b*a>;
// generators/relations
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