direct product, abelian, monomial, 2-elementary
Aliases: C4×C88, SmallGroup(352,45)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C4×C88 |
| C1 — C4×C88 |
| C1 — C4×C88 |
Generators and relations for C4×C88
G = < a,b | a4=b88=1, ab=ba >
Smallest permutation representation of C4×C88
►Regular action on 352 points
Generators in S352
(1 136 202 343)(2 137 203 344)(3 138 204 345)(4 139 205 346)(5 140 206 347)(6 141 207 348)(7 142 208 349)(8 143 209 350)(9 144 210 351)(10 145 211 352)(11 146 212 265)(12 147 213 266)(13 148 214 267)(14 149 215 268)(15 150 216 269)(16 151 217 270)(17 152 218 271)(18 153 219 272)(19 154 220 273)(20 155 221 274)(21 156 222 275)(22 157 223 276)(23 158 224 277)(24 159 225 278)(25 160 226 279)(26 161 227 280)(27 162 228 281)(28 163 229 282)(29 164 230 283)(30 165 231 284)(31 166 232 285)(32 167 233 286)(33 168 234 287)(34 169 235 288)(35 170 236 289)(36 171 237 290)(37 172 238 291)(38 173 239 292)(39 174 240 293)(40 175 241 294)(41 176 242 295)(42 89 243 296)(43 90 244 297)(44 91 245 298)(45 92 246 299)(46 93 247 300)(47 94 248 301)(48 95 249 302)(49 96 250 303)(50 97 251 304)(51 98 252 305)(52 99 253 306)(53 100 254 307)(54 101 255 308)(55 102 256 309)(56 103 257 310)(57 104 258 311)(58 105 259 312)(59 106 260 313)(60 107 261 314)(61 108 262 315)(62 109 263 316)(63 110 264 317)(64 111 177 318)(65 112 178 319)(66 113 179 320)(67 114 180 321)(68 115 181 322)(69 116 182 323)(70 117 183 324)(71 118 184 325)(72 119 185 326)(73 120 186 327)(74 121 187 328)(75 122 188 329)(76 123 189 330)(77 124 190 331)(78 125 191 332)(79 126 192 333)(80 127 193 334)(81 128 194 335)(82 129 195 336)(83 130 196 337)(84 131 197 338)(85 132 198 339)(86 133 199 340)(87 134 200 341)(88 135 201 342)
(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)(265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352)
G:=sub<Sym(352)| (1,136,202,343)(2,137,203,344)(3,138,204,345)(4,139,205,346)(5,140,206,347)(6,141,207,348)(7,142,208,349)(8,143,209,350)(9,144,210,351)(10,145,211,352)(11,146,212,265)(12,147,213,266)(13,148,214,267)(14,149,215,268)(15,150,216,269)(16,151,217,270)(17,152,218,271)(18,153,219,272)(19,154,220,273)(20,155,221,274)(21,156,222,275)(22,157,223,276)(23,158,224,277)(24,159,225,278)(25,160,226,279)(26,161,227,280)(27,162,228,281)(28,163,229,282)(29,164,230,283)(30,165,231,284)(31,166,232,285)(32,167,233,286)(33,168,234,287)(34,169,235,288)(35,170,236,289)(36,171,237,290)(37,172,238,291)(38,173,239,292)(39,174,240,293)(40,175,241,294)(41,176,242,295)(42,89,243,296)(43,90,244,297)(44,91,245,298)(45,92,246,299)(46,93,247,300)(47,94,248,301)(48,95,249,302)(49,96,250,303)(50,97,251,304)(51,98,252,305)(52,99,253,306)(53,100,254,307)(54,101,255,308)(55,102,256,309)(56,103,257,310)(57,104,258,311)(58,105,259,312)(59,106,260,313)(60,107,261,314)(61,108,262,315)(62,109,263,316)(63,110,264,317)(64,111,177,318)(65,112,178,319)(66,113,179,320)(67,114,180,321)(68,115,181,322)(69,116,182,323)(70,117,183,324)(71,118,184,325)(72,119,185,326)(73,120,186,327)(74,121,187,328)(75,122,188,329)(76,123,189,330)(77,124,190,331)(78,125,191,332)(79,126,192,333)(80,127,193,334)(81,128,194,335)(82,129,195,336)(83,130,196,337)(84,131,197,338)(85,132,198,339)(86,133,199,340)(87,134,200,341)(88,135,201,342), (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)(265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352)>;
G:=Group( (1,136,202,343)(2,137,203,344)(3,138,204,345)(4,139,205,346)(5,140,206,347)(6,141,207,348)(7,142,208,349)(8,143,209,350)(9,144,210,351)(10,145,211,352)(11,146,212,265)(12,147,213,266)(13,148,214,267)(14,149,215,268)(15,150,216,269)(16,151,217,270)(17,152,218,271)(18,153,219,272)(19,154,220,273)(20,155,221,274)(21,156,222,275)(22,157,223,276)(23,158,224,277)(24,159,225,278)(25,160,226,279)(26,161,227,280)(27,162,228,281)(28,163,229,282)(29,164,230,283)(30,165,231,284)(31,166,232,285)(32,167,233,286)(33,168,234,287)(34,169,235,288)(35,170,236,289)(36,171,237,290)(37,172,238,291)(38,173,239,292)(39,174,240,293)(40,175,241,294)(41,176,242,295)(42,89,243,296)(43,90,244,297)(44,91,245,298)(45,92,246,299)(46,93,247,300)(47,94,248,301)(48,95,249,302)(49,96,250,303)(50,97,251,304)(51,98,252,305)(52,99,253,306)(53,100,254,307)(54,101,255,308)(55,102,256,309)(56,103,257,310)(57,104,258,311)(58,105,259,312)(59,106,260,313)(60,107,261,314)(61,108,262,315)(62,109,263,316)(63,110,264,317)(64,111,177,318)(65,112,178,319)(66,113,179,320)(67,114,180,321)(68,115,181,322)(69,116,182,323)(70,117,183,324)(71,118,184,325)(72,119,185,326)(73,120,186,327)(74,121,187,328)(75,122,188,329)(76,123,189,330)(77,124,190,331)(78,125,191,332)(79,126,192,333)(80,127,193,334)(81,128,194,335)(82,129,195,336)(83,130,196,337)(84,131,197,338)(85,132,198,339)(86,133,199,340)(87,134,200,341)(88,135,201,342), (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)(265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352) );
G=PermutationGroup([[(1,136,202,343),(2,137,203,344),(3,138,204,345),(4,139,205,346),(5,140,206,347),(6,141,207,348),(7,142,208,349),(8,143,209,350),(9,144,210,351),(10,145,211,352),(11,146,212,265),(12,147,213,266),(13,148,214,267),(14,149,215,268),(15,150,216,269),(16,151,217,270),(17,152,218,271),(18,153,219,272),(19,154,220,273),(20,155,221,274),(21,156,222,275),(22,157,223,276),(23,158,224,277),(24,159,225,278),(25,160,226,279),(26,161,227,280),(27,162,228,281),(28,163,229,282),(29,164,230,283),(30,165,231,284),(31,166,232,285),(32,167,233,286),(33,168,234,287),(34,169,235,288),(35,170,236,289),(36,171,237,290),(37,172,238,291),(38,173,239,292),(39,174,240,293),(40,175,241,294),(41,176,242,295),(42,89,243,296),(43,90,244,297),(44,91,245,298),(45,92,246,299),(46,93,247,300),(47,94,248,301),(48,95,249,302),(49,96,250,303),(50,97,251,304),(51,98,252,305),(52,99,253,306),(53,100,254,307),(54,101,255,308),(55,102,256,309),(56,103,257,310),(57,104,258,311),(58,105,259,312),(59,106,260,313),(60,107,261,314),(61,108,262,315),(62,109,263,316),(63,110,264,317),(64,111,177,318),(65,112,178,319),(66,113,179,320),(67,114,180,321),(68,115,181,322),(69,116,182,323),(70,117,183,324),(71,118,184,325),(72,119,185,326),(73,120,186,327),(74,121,187,328),(75,122,188,329),(76,123,189,330),(77,124,190,331),(78,125,191,332),(79,126,192,333),(80,127,193,334),(81,128,194,335),(82,129,195,336),(83,130,196,337),(84,131,197,338),(85,132,198,339),(86,133,199,340),(87,134,200,341),(88,135,201,342)], [(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),(265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352)]])
352 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 8A | ··· | 8P | 11A | ··· | 11J | 22A | ··· | 22AD | 44A | ··· | 44DP | 88A | ··· | 88FD |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
352 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C11 | C22 | C22 | C44 | C44 | C88 |
| kernel | C4×C88 | C4×C44 | C2×C88 | C88 | C2×C44 | C44 | C4×C8 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 8 | 4 | 16 | 10 | 10 | 20 | 80 | 40 | 160 |
Matrix representation of C4×C88 ►in GL2(𝔽89) generated by
| 55 | 0 |
| 0 | 55 |
| 4 | 0 |
| 0 | 76 |
G:=sub<GL(2,GF(89))| [55,0,0,55],[4,0,0,76] >;
C4×C88 in GAP, Magma, Sage, TeX
C_4\times C_{88} % in TeX
G:=Group("C4xC88"); // GroupNames label
G:=SmallGroup(352,45);
// by ID
G=gap.SmallGroup(352,45);
# by ID
G:=PCGroup([6,-2,-2,-11,-2,-2,-2,264,535,117]);
// Polycyclic
G:=Group<a,b|a^4=b^88=1,a*b=b*a>;
// generators/relations
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