direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C44⋊C4, C22.15D44, C23.30D22, C22.5Dic22, C44⋊7(C2×C4), (C2×C44)⋊5C4, C22⋊2(C4⋊C4), C2.2(C2×D44), C22.9(C2×Q8), (C2×C22).6Q8, C4⋊2(C2×Dic11), (C2×C4)⋊3Dic11, C22.15(C2×D4), (C2×C22).20D4, (C2×C4).84D22, (C22×C44).7C2, C2.3(C2×Dic22), (C22×C4).6D11, C22.23(C22×C4), (C2×C22).43C23, (C2×C44).92C22, C2.4(C22×Dic11), (C22×C22).35C22, (C22×Dic11).5C2, C22.14(C2×Dic11), C22.21(C22×D11), (C2×Dic11).34C22, C11⋊3(C2×C4⋊C4), (C2×C22).34(C2×C4), SmallGroup(352,120)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C44⋊C4
G = < a,b,c | a2=b44=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 378 in 92 conjugacy classes, 65 normal (15 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C11, C4⋊C4, C22×C4, C22×C4, C22, C22, C2×C4⋊C4, Dic11, C44, C2×C22, C2×C22, C2×Dic11, C2×Dic11, C2×C44, C22×C22, C44⋊C4, C22×Dic11, C22×C44, C2×C44⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, D11, C2×C4⋊C4, Dic11, D22, Dic22, D44, C2×Dic11, C22×D11, C44⋊C4, C2×Dic22, C2×D44, C22×Dic11, C2×C44⋊C4
►Regular action on 352 points
Generators in S352
(1 266)(2 267)(3 268)(4 269)(5 270)(6 271)(7 272)(8 273)(9 274)(10 275)(11 276)(12 277)(13 278)(14 279)(15 280)(16 281)(17 282)(18 283)(19 284)(20 285)(21 286)(22 287)(23 288)(24 289)(25 290)(26 291)(27 292)(28 293)(29 294)(30 295)(31 296)(32 297)(33 298)(34 299)(35 300)(36 301)(37 302)(38 303)(39 304)(40 305)(41 306)(42 307)(43 308)(44 265)(45 159)(46 160)(47 161)(48 162)(49 163)(50 164)(51 165)(52 166)(53 167)(54 168)(55 169)(56 170)(57 171)(58 172)(59 173)(60 174)(61 175)(62 176)(63 133)(64 134)(65 135)(66 136)(67 137)(68 138)(69 139)(70 140)(71 141)(72 142)(73 143)(74 144)(75 145)(76 146)(77 147)(78 148)(79 149)(80 150)(81 151)(82 152)(83 153)(84 154)(85 155)(86 156)(87 157)(88 158)(89 219)(90 220)(91 177)(92 178)(93 179)(94 180)(95 181)(96 182)(97 183)(98 184)(99 185)(100 186)(101 187)(102 188)(103 189)(104 190)(105 191)(106 192)(107 193)(108 194)(109 195)(110 196)(111 197)(112 198)(113 199)(114 200)(115 201)(116 202)(117 203)(118 204)(119 205)(120 206)(121 207)(122 208)(123 209)(124 210)(125 211)(126 212)(127 213)(128 214)(129 215)(130 216)(131 217)(132 218)(221 338)(222 339)(223 340)(224 341)(225 342)(226 343)(227 344)(228 345)(229 346)(230 347)(231 348)(232 349)(233 350)(234 351)(235 352)(236 309)(237 310)(238 311)(239 312)(240 313)(241 314)(242 315)(243 316)(244 317)(245 318)(246 319)(247 320)(248 321)(249 322)(250 323)(251 324)(252 325)(253 326)(254 327)(255 328)(256 329)(257 330)(258 331)(259 332)(260 333)(261 334)(262 335)(263 336)(264 337)
(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)
(1 243 136 212)(2 242 137 211)(3 241 138 210)(4 240 139 209)(5 239 140 208)(6 238 141 207)(7 237 142 206)(8 236 143 205)(9 235 144 204)(10 234 145 203)(11 233 146 202)(12 232 147 201)(13 231 148 200)(14 230 149 199)(15 229 150 198)(16 228 151 197)(17 227 152 196)(18 226 153 195)(19 225 154 194)(20 224 155 193)(21 223 156 192)(22 222 157 191)(23 221 158 190)(24 264 159 189)(25 263 160 188)(26 262 161 187)(27 261 162 186)(28 260 163 185)(29 259 164 184)(30 258 165 183)(31 257 166 182)(32 256 167 181)(33 255 168 180)(34 254 169 179)(35 253 170 178)(36 252 171 177)(37 251 172 220)(38 250 173 219)(39 249 174 218)(40 248 175 217)(41 247 176 216)(42 246 133 215)(43 245 134 214)(44 244 135 213)(45 103 289 337)(46 102 290 336)(47 101 291 335)(48 100 292 334)(49 99 293 333)(50 98 294 332)(51 97 295 331)(52 96 296 330)(53 95 297 329)(54 94 298 328)(55 93 299 327)(56 92 300 326)(57 91 301 325)(58 90 302 324)(59 89 303 323)(60 132 304 322)(61 131 305 321)(62 130 306 320)(63 129 307 319)(64 128 308 318)(65 127 265 317)(66 126 266 316)(67 125 267 315)(68 124 268 314)(69 123 269 313)(70 122 270 312)(71 121 271 311)(72 120 272 310)(73 119 273 309)(74 118 274 352)(75 117 275 351)(76 116 276 350)(77 115 277 349)(78 114 278 348)(79 113 279 347)(80 112 280 346)(81 111 281 345)(82 110 282 344)(83 109 283 343)(84 108 284 342)(85 107 285 341)(86 106 286 340)(87 105 287 339)(88 104 288 338)
G:=sub<Sym(352)| (1,266)(2,267)(3,268)(4,269)(5,270)(6,271)(7,272)(8,273)(9,274)(10,275)(11,276)(12,277)(13,278)(14,279)(15,280)(16,281)(17,282)(18,283)(19,284)(20,285)(21,286)(22,287)(23,288)(24,289)(25,290)(26,291)(27,292)(28,293)(29,294)(30,295)(31,296)(32,297)(33,298)(34,299)(35,300)(36,301)(37,302)(38,303)(39,304)(40,305)(41,306)(42,307)(43,308)(44,265)(45,159)(46,160)(47,161)(48,162)(49,163)(50,164)(51,165)(52,166)(53,167)(54,168)(55,169)(56,170)(57,171)(58,172)(59,173)(60,174)(61,175)(62,176)(63,133)(64,134)(65,135)(66,136)(67,137)(68,138)(69,139)(70,140)(71,141)(72,142)(73,143)(74,144)(75,145)(76,146)(77,147)(78,148)(79,149)(80,150)(81,151)(82,152)(83,153)(84,154)(85,155)(86,156)(87,157)(88,158)(89,219)(90,220)(91,177)(92,178)(93,179)(94,180)(95,181)(96,182)(97,183)(98,184)(99,185)(100,186)(101,187)(102,188)(103,189)(104,190)(105,191)(106,192)(107,193)(108,194)(109,195)(110,196)(111,197)(112,198)(113,199)(114,200)(115,201)(116,202)(117,203)(118,204)(119,205)(120,206)(121,207)(122,208)(123,209)(124,210)(125,211)(126,212)(127,213)(128,214)(129,215)(130,216)(131,217)(132,218)(221,338)(222,339)(223,340)(224,341)(225,342)(226,343)(227,344)(228,345)(229,346)(230,347)(231,348)(232,349)(233,350)(234,351)(235,352)(236,309)(237,310)(238,311)(239,312)(240,313)(241,314)(242,315)(243,316)(244,317)(245,318)(246,319)(247,320)(248,321)(249,322)(250,323)(251,324)(252,325)(253,326)(254,327)(255,328)(256,329)(257,330)(258,331)(259,332)(260,333)(261,334)(262,335)(263,336)(264,337), (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), (1,243,136,212)(2,242,137,211)(3,241,138,210)(4,240,139,209)(5,239,140,208)(6,238,141,207)(7,237,142,206)(8,236,143,205)(9,235,144,204)(10,234,145,203)(11,233,146,202)(12,232,147,201)(13,231,148,200)(14,230,149,199)(15,229,150,198)(16,228,151,197)(17,227,152,196)(18,226,153,195)(19,225,154,194)(20,224,155,193)(21,223,156,192)(22,222,157,191)(23,221,158,190)(24,264,159,189)(25,263,160,188)(26,262,161,187)(27,261,162,186)(28,260,163,185)(29,259,164,184)(30,258,165,183)(31,257,166,182)(32,256,167,181)(33,255,168,180)(34,254,169,179)(35,253,170,178)(36,252,171,177)(37,251,172,220)(38,250,173,219)(39,249,174,218)(40,248,175,217)(41,247,176,216)(42,246,133,215)(43,245,134,214)(44,244,135,213)(45,103,289,337)(46,102,290,336)(47,101,291,335)(48,100,292,334)(49,99,293,333)(50,98,294,332)(51,97,295,331)(52,96,296,330)(53,95,297,329)(54,94,298,328)(55,93,299,327)(56,92,300,326)(57,91,301,325)(58,90,302,324)(59,89,303,323)(60,132,304,322)(61,131,305,321)(62,130,306,320)(63,129,307,319)(64,128,308,318)(65,127,265,317)(66,126,266,316)(67,125,267,315)(68,124,268,314)(69,123,269,313)(70,122,270,312)(71,121,271,311)(72,120,272,310)(73,119,273,309)(74,118,274,352)(75,117,275,351)(76,116,276,350)(77,115,277,349)(78,114,278,348)(79,113,279,347)(80,112,280,346)(81,111,281,345)(82,110,282,344)(83,109,283,343)(84,108,284,342)(85,107,285,341)(86,106,286,340)(87,105,287,339)(88,104,288,338)>;
G:=Group( (1,266)(2,267)(3,268)(4,269)(5,270)(6,271)(7,272)(8,273)(9,274)(10,275)(11,276)(12,277)(13,278)(14,279)(15,280)(16,281)(17,282)(18,283)(19,284)(20,285)(21,286)(22,287)(23,288)(24,289)(25,290)(26,291)(27,292)(28,293)(29,294)(30,295)(31,296)(32,297)(33,298)(34,299)(35,300)(36,301)(37,302)(38,303)(39,304)(40,305)(41,306)(42,307)(43,308)(44,265)(45,159)(46,160)(47,161)(48,162)(49,163)(50,164)(51,165)(52,166)(53,167)(54,168)(55,169)(56,170)(57,171)(58,172)(59,173)(60,174)(61,175)(62,176)(63,133)(64,134)(65,135)(66,136)(67,137)(68,138)(69,139)(70,140)(71,141)(72,142)(73,143)(74,144)(75,145)(76,146)(77,147)(78,148)(79,149)(80,150)(81,151)(82,152)(83,153)(84,154)(85,155)(86,156)(87,157)(88,158)(89,219)(90,220)(91,177)(92,178)(93,179)(94,180)(95,181)(96,182)(97,183)(98,184)(99,185)(100,186)(101,187)(102,188)(103,189)(104,190)(105,191)(106,192)(107,193)(108,194)(109,195)(110,196)(111,197)(112,198)(113,199)(114,200)(115,201)(116,202)(117,203)(118,204)(119,205)(120,206)(121,207)(122,208)(123,209)(124,210)(125,211)(126,212)(127,213)(128,214)(129,215)(130,216)(131,217)(132,218)(221,338)(222,339)(223,340)(224,341)(225,342)(226,343)(227,344)(228,345)(229,346)(230,347)(231,348)(232,349)(233,350)(234,351)(235,352)(236,309)(237,310)(238,311)(239,312)(240,313)(241,314)(242,315)(243,316)(244,317)(245,318)(246,319)(247,320)(248,321)(249,322)(250,323)(251,324)(252,325)(253,326)(254,327)(255,328)(256,329)(257,330)(258,331)(259,332)(260,333)(261,334)(262,335)(263,336)(264,337), (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), (1,243,136,212)(2,242,137,211)(3,241,138,210)(4,240,139,209)(5,239,140,208)(6,238,141,207)(7,237,142,206)(8,236,143,205)(9,235,144,204)(10,234,145,203)(11,233,146,202)(12,232,147,201)(13,231,148,200)(14,230,149,199)(15,229,150,198)(16,228,151,197)(17,227,152,196)(18,226,153,195)(19,225,154,194)(20,224,155,193)(21,223,156,192)(22,222,157,191)(23,221,158,190)(24,264,159,189)(25,263,160,188)(26,262,161,187)(27,261,162,186)(28,260,163,185)(29,259,164,184)(30,258,165,183)(31,257,166,182)(32,256,167,181)(33,255,168,180)(34,254,169,179)(35,253,170,178)(36,252,171,177)(37,251,172,220)(38,250,173,219)(39,249,174,218)(40,248,175,217)(41,247,176,216)(42,246,133,215)(43,245,134,214)(44,244,135,213)(45,103,289,337)(46,102,290,336)(47,101,291,335)(48,100,292,334)(49,99,293,333)(50,98,294,332)(51,97,295,331)(52,96,296,330)(53,95,297,329)(54,94,298,328)(55,93,299,327)(56,92,300,326)(57,91,301,325)(58,90,302,324)(59,89,303,323)(60,132,304,322)(61,131,305,321)(62,130,306,320)(63,129,307,319)(64,128,308,318)(65,127,265,317)(66,126,266,316)(67,125,267,315)(68,124,268,314)(69,123,269,313)(70,122,270,312)(71,121,271,311)(72,120,272,310)(73,119,273,309)(74,118,274,352)(75,117,275,351)(76,116,276,350)(77,115,277,349)(78,114,278,348)(79,113,279,347)(80,112,280,346)(81,111,281,345)(82,110,282,344)(83,109,283,343)(84,108,284,342)(85,107,285,341)(86,106,286,340)(87,105,287,339)(88,104,288,338) );
G=PermutationGroup([[(1,266),(2,267),(3,268),(4,269),(5,270),(6,271),(7,272),(8,273),(9,274),(10,275),(11,276),(12,277),(13,278),(14,279),(15,280),(16,281),(17,282),(18,283),(19,284),(20,285),(21,286),(22,287),(23,288),(24,289),(25,290),(26,291),(27,292),(28,293),(29,294),(30,295),(31,296),(32,297),(33,298),(34,299),(35,300),(36,301),(37,302),(38,303),(39,304),(40,305),(41,306),(42,307),(43,308),(44,265),(45,159),(46,160),(47,161),(48,162),(49,163),(50,164),(51,165),(52,166),(53,167),(54,168),(55,169),(56,170),(57,171),(58,172),(59,173),(60,174),(61,175),(62,176),(63,133),(64,134),(65,135),(66,136),(67,137),(68,138),(69,139),(70,140),(71,141),(72,142),(73,143),(74,144),(75,145),(76,146),(77,147),(78,148),(79,149),(80,150),(81,151),(82,152),(83,153),(84,154),(85,155),(86,156),(87,157),(88,158),(89,219),(90,220),(91,177),(92,178),(93,179),(94,180),(95,181),(96,182),(97,183),(98,184),(99,185),(100,186),(101,187),(102,188),(103,189),(104,190),(105,191),(106,192),(107,193),(108,194),(109,195),(110,196),(111,197),(112,198),(113,199),(114,200),(115,201),(116,202),(117,203),(118,204),(119,205),(120,206),(121,207),(122,208),(123,209),(124,210),(125,211),(126,212),(127,213),(128,214),(129,215),(130,216),(131,217),(132,218),(221,338),(222,339),(223,340),(224,341),(225,342),(226,343),(227,344),(228,345),(229,346),(230,347),(231,348),(232,349),(233,350),(234,351),(235,352),(236,309),(237,310),(238,311),(239,312),(240,313),(241,314),(242,315),(243,316),(244,317),(245,318),(246,319),(247,320),(248,321),(249,322),(250,323),(251,324),(252,325),(253,326),(254,327),(255,328),(256,329),(257,330),(258,331),(259,332),(260,333),(261,334),(262,335),(263,336),(264,337)], [(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)], [(1,243,136,212),(2,242,137,211),(3,241,138,210),(4,240,139,209),(5,239,140,208),(6,238,141,207),(7,237,142,206),(8,236,143,205),(9,235,144,204),(10,234,145,203),(11,233,146,202),(12,232,147,201),(13,231,148,200),(14,230,149,199),(15,229,150,198),(16,228,151,197),(17,227,152,196),(18,226,153,195),(19,225,154,194),(20,224,155,193),(21,223,156,192),(22,222,157,191),(23,221,158,190),(24,264,159,189),(25,263,160,188),(26,262,161,187),(27,261,162,186),(28,260,163,185),(29,259,164,184),(30,258,165,183),(31,257,166,182),(32,256,167,181),(33,255,168,180),(34,254,169,179),(35,253,170,178),(36,252,171,177),(37,251,172,220),(38,250,173,219),(39,249,174,218),(40,248,175,217),(41,247,176,216),(42,246,133,215),(43,245,134,214),(44,244,135,213),(45,103,289,337),(46,102,290,336),(47,101,291,335),(48,100,292,334),(49,99,293,333),(50,98,294,332),(51,97,295,331),(52,96,296,330),(53,95,297,329),(54,94,298,328),(55,93,299,327),(56,92,300,326),(57,91,301,325),(58,90,302,324),(59,89,303,323),(60,132,304,322),(61,131,305,321),(62,130,306,320),(63,129,307,319),(64,128,308,318),(65,127,265,317),(66,126,266,316),(67,125,267,315),(68,124,268,314),(69,123,269,313),(70,122,270,312),(71,121,271,311),(72,120,272,310),(73,119,273,309),(74,118,274,352),(75,117,275,351),(76,116,276,350),(77,115,277,349),(78,114,278,348),(79,113,279,347),(80,112,280,346),(81,111,281,345),(82,110,282,344),(83,109,283,343),(84,108,284,342),(85,107,285,341),(86,106,286,340),(87,105,287,339),(88,104,288,338)]])
100 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 11A | ··· | 11E | 22A | ··· | 22AI | 44A | ··· | 44AN |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 22 | ··· | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C4 | D4 | Q8 | D11 | Dic11 | D22 | D22 | Dic22 | D44 |
| kernel | C2×C44⋊C4 | C44⋊C4 | C22×Dic11 | C22×C44 | C2×C44 | C2×C22 | C2×C22 | C22×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 2 | 1 | 8 | 2 | 2 | 5 | 20 | 10 | 5 | 20 | 20 |
Matrix representation of C2×C44⋊C4 ►in GL5(𝔽89)
| 88 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 0 | 88 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 44 | 17 | 0 | 0 |
| 0 | 72 | 42 | 0 | 0 |
| 0 | 0 | 0 | 87 | 54 |
| 0 | 0 | 0 | 59 | 53 |
| 88 | 0 | 0 | 0 | 0 |
| 0 | 13 | 26 | 0 | 0 |
| 0 | 14 | 76 | 0 | 0 |
| 0 | 0 | 0 | 41 | 1 |
| 0 | 0 | 0 | 11 | 48 |
G:=sub<GL(5,GF(89))| [88,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,88,0,0,0,0,0,88],[1,0,0,0,0,0,44,72,0,0,0,17,42,0,0,0,0,0,87,59,0,0,0,54,53],[88,0,0,0,0,0,13,14,0,0,0,26,76,0,0,0,0,0,41,11,0,0,0,1,48] >;
C2×C44⋊C4 in GAP, Magma, Sage, TeX
C_2\times C_{44}\rtimes C_4 % in TeX
G:=Group("C2xC44:C4"); // GroupNames label
G:=SmallGroup(352,120);
// by ID
G=gap.SmallGroup(352,120);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,48,362,86,11525]);
// Polycyclic
G:=Group<a,b,c|a^2=b^44=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations