direct product, abelian, monomial, 2-elementary
Aliases: C2×C152, SmallGroup(304,22)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C152 |
| C1 — C2×C152 |
| C1 — C2×C152 |
Generators and relations for C2×C152
G = < a,b | a2=b152=1, ab=ba >
Smallest permutation representation of C2×C152
►Regular action on 304 points
Generators in S304
(1 269)(2 270)(3 271)(4 272)(5 273)(6 274)(7 275)(8 276)(9 277)(10 278)(11 279)(12 280)(13 281)(14 282)(15 283)(16 284)(17 285)(18 286)(19 287)(20 288)(21 289)(22 290)(23 291)(24 292)(25 293)(26 294)(27 295)(28 296)(29 297)(30 298)(31 299)(32 300)(33 301)(34 302)(35 303)(36 304)(37 153)(38 154)(39 155)(40 156)(41 157)(42 158)(43 159)(44 160)(45 161)(46 162)(47 163)(48 164)(49 165)(50 166)(51 167)(52 168)(53 169)(54 170)(55 171)(56 172)(57 173)(58 174)(59 175)(60 176)(61 177)(62 178)(63 179)(64 180)(65 181)(66 182)(67 183)(68 184)(69 185)(70 186)(71 187)(72 188)(73 189)(74 190)(75 191)(76 192)(77 193)(78 194)(79 195)(80 196)(81 197)(82 198)(83 199)(84 200)(85 201)(86 202)(87 203)(88 204)(89 205)(90 206)(91 207)(92 208)(93 209)(94 210)(95 211)(96 212)(97 213)(98 214)(99 215)(100 216)(101 217)(102 218)(103 219)(104 220)(105 221)(106 222)(107 223)(108 224)(109 225)(110 226)(111 227)(112 228)(113 229)(114 230)(115 231)(116 232)(117 233)(118 234)(119 235)(120 236)(121 237)(122 238)(123 239)(124 240)(125 241)(126 242)(127 243)(128 244)(129 245)(130 246)(131 247)(132 248)(133 249)(134 250)(135 251)(136 252)(137 253)(138 254)(139 255)(140 256)(141 257)(142 258)(143 259)(144 260)(145 261)(146 262)(147 263)(148 264)(149 265)(150 266)(151 267)(152 268)
(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)(153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304)
G:=sub<Sym(304)| (1,269)(2,270)(3,271)(4,272)(5,273)(6,274)(7,275)(8,276)(9,277)(10,278)(11,279)(12,280)(13,281)(14,282)(15,283)(16,284)(17,285)(18,286)(19,287)(20,288)(21,289)(22,290)(23,291)(24,292)(25,293)(26,294)(27,295)(28,296)(29,297)(30,298)(31,299)(32,300)(33,301)(34,302)(35,303)(36,304)(37,153)(38,154)(39,155)(40,156)(41,157)(42,158)(43,159)(44,160)(45,161)(46,162)(47,163)(48,164)(49,165)(50,166)(51,167)(52,168)(53,169)(54,170)(55,171)(56,172)(57,173)(58,174)(59,175)(60,176)(61,177)(62,178)(63,179)(64,180)(65,181)(66,182)(67,183)(68,184)(69,185)(70,186)(71,187)(72,188)(73,189)(74,190)(75,191)(76,192)(77,193)(78,194)(79,195)(80,196)(81,197)(82,198)(83,199)(84,200)(85,201)(86,202)(87,203)(88,204)(89,205)(90,206)(91,207)(92,208)(93,209)(94,210)(95,211)(96,212)(97,213)(98,214)(99,215)(100,216)(101,217)(102,218)(103,219)(104,220)(105,221)(106,222)(107,223)(108,224)(109,225)(110,226)(111,227)(112,228)(113,229)(114,230)(115,231)(116,232)(117,233)(118,234)(119,235)(120,236)(121,237)(122,238)(123,239)(124,240)(125,241)(126,242)(127,243)(128,244)(129,245)(130,246)(131,247)(132,248)(133,249)(134,250)(135,251)(136,252)(137,253)(138,254)(139,255)(140,256)(141,257)(142,258)(143,259)(144,260)(145,261)(146,262)(147,263)(148,264)(149,265)(150,266)(151,267)(152,268), (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)(153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304)>;
G:=Group( (1,269)(2,270)(3,271)(4,272)(5,273)(6,274)(7,275)(8,276)(9,277)(10,278)(11,279)(12,280)(13,281)(14,282)(15,283)(16,284)(17,285)(18,286)(19,287)(20,288)(21,289)(22,290)(23,291)(24,292)(25,293)(26,294)(27,295)(28,296)(29,297)(30,298)(31,299)(32,300)(33,301)(34,302)(35,303)(36,304)(37,153)(38,154)(39,155)(40,156)(41,157)(42,158)(43,159)(44,160)(45,161)(46,162)(47,163)(48,164)(49,165)(50,166)(51,167)(52,168)(53,169)(54,170)(55,171)(56,172)(57,173)(58,174)(59,175)(60,176)(61,177)(62,178)(63,179)(64,180)(65,181)(66,182)(67,183)(68,184)(69,185)(70,186)(71,187)(72,188)(73,189)(74,190)(75,191)(76,192)(77,193)(78,194)(79,195)(80,196)(81,197)(82,198)(83,199)(84,200)(85,201)(86,202)(87,203)(88,204)(89,205)(90,206)(91,207)(92,208)(93,209)(94,210)(95,211)(96,212)(97,213)(98,214)(99,215)(100,216)(101,217)(102,218)(103,219)(104,220)(105,221)(106,222)(107,223)(108,224)(109,225)(110,226)(111,227)(112,228)(113,229)(114,230)(115,231)(116,232)(117,233)(118,234)(119,235)(120,236)(121,237)(122,238)(123,239)(124,240)(125,241)(126,242)(127,243)(128,244)(129,245)(130,246)(131,247)(132,248)(133,249)(134,250)(135,251)(136,252)(137,253)(138,254)(139,255)(140,256)(141,257)(142,258)(143,259)(144,260)(145,261)(146,262)(147,263)(148,264)(149,265)(150,266)(151,267)(152,268), (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)(153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304) );
G=PermutationGroup([[(1,269),(2,270),(3,271),(4,272),(5,273),(6,274),(7,275),(8,276),(9,277),(10,278),(11,279),(12,280),(13,281),(14,282),(15,283),(16,284),(17,285),(18,286),(19,287),(20,288),(21,289),(22,290),(23,291),(24,292),(25,293),(26,294),(27,295),(28,296),(29,297),(30,298),(31,299),(32,300),(33,301),(34,302),(35,303),(36,304),(37,153),(38,154),(39,155),(40,156),(41,157),(42,158),(43,159),(44,160),(45,161),(46,162),(47,163),(48,164),(49,165),(50,166),(51,167),(52,168),(53,169),(54,170),(55,171),(56,172),(57,173),(58,174),(59,175),(60,176),(61,177),(62,178),(63,179),(64,180),(65,181),(66,182),(67,183),(68,184),(69,185),(70,186),(71,187),(72,188),(73,189),(74,190),(75,191),(76,192),(77,193),(78,194),(79,195),(80,196),(81,197),(82,198),(83,199),(84,200),(85,201),(86,202),(87,203),(88,204),(89,205),(90,206),(91,207),(92,208),(93,209),(94,210),(95,211),(96,212),(97,213),(98,214),(99,215),(100,216),(101,217),(102,218),(103,219),(104,220),(105,221),(106,222),(107,223),(108,224),(109,225),(110,226),(111,227),(112,228),(113,229),(114,230),(115,231),(116,232),(117,233),(118,234),(119,235),(120,236),(121,237),(122,238),(123,239),(124,240),(125,241),(126,242),(127,243),(128,244),(129,245),(130,246),(131,247),(132,248),(133,249),(134,250),(135,251),(136,252),(137,253),(138,254),(139,255),(140,256),(141,257),(142,258),(143,259),(144,260),(145,261),(146,262),(147,263),(148,264),(149,265),(150,266),(151,267),(152,268)], [(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),(153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304)]])
304 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | ··· | 8H | 19A | ··· | 19R | 38A | ··· | 38BB | 76A | ··· | 76BT | 152A | ··· | 152EN |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 | 152 | ··· | 152 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
304 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C19 | C38 | C38 | C76 | C76 | C152 |
| kernel | C2×C152 | C152 | C2×C76 | C76 | C2×C38 | C38 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 18 | 36 | 18 | 36 | 36 | 144 |
Matrix representation of C2×C152 ►in GL2(𝔽457) generated by
| 456 | 0 |
| 0 | 1 |
| 446 | 0 |
| 0 | 172 |
G:=sub<GL(2,GF(457))| [456,0,0,1],[446,0,0,172] >;
C2×C152 in GAP, Magma, Sage, TeX
C_2\times C_{152} % in TeX
G:=Group("C2xC152"); // GroupNames label
G:=SmallGroup(304,22);
// by ID
G=gap.SmallGroup(304,22);
# by ID
G:=PCGroup([5,-2,-2,-19,-2,-2,380,58]);
// Polycyclic
G:=Group<a,b|a^2=b^152=1,a*b=b*a>;
// generators/relations
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