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## G = C2×C4×C44order 352 = 25·11

### Abelian group of type [2,4,44]

Aliases: C2×C4×C44, SmallGroup(352,149)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4×C44
 Chief series C1 — C2 — C22 — C2×C22 — C2×C44 — C4×C44 — C2×C4×C44
 Lower central C1 — C2×C4×C44
 Upper central C1 — C2×C4×C44

Generators and relations for C2×C4×C44
G = < a,b,c | a2=b4=c44=1, ab=ba, ac=ca, bc=cb >

Subgroups: 108, all normal (8 characteristic)
C1, C2, C4, C22, C22, C2×C4, C23, C11, C42, C22×C4, C22, C2×C42, C44, C2×C22, C2×C22, C2×C44, C22×C22, C4×C44, C22×C44, C2×C4×C44
Quotients: C1, C2, C4, C22, C2×C4, C23, C11, C42, C22×C4, C22, C2×C42, C44, C2×C22, C2×C44, C22×C22, C4×C44, C22×C44, C2×C4×C44

Smallest permutation representation of C2×C4×C44
Regular action on 352 points
Generators in S352
(1 124)(2 125)(3 126)(4 127)(5 128)(6 129)(7 130)(8 131)(9 132)(10 89)(11 90)(12 91)(13 92)(14 93)(15 94)(16 95)(17 96)(18 97)(19 98)(20 99)(21 100)(22 101)(23 102)(24 103)(25 104)(26 105)(27 106)(28 107)(29 108)(30 109)(31 110)(32 111)(33 112)(34 113)(35 114)(36 115)(37 116)(38 117)(39 118)(40 119)(41 120)(42 121)(43 122)(44 123)(45 134)(46 135)(47 136)(48 137)(49 138)(50 139)(51 140)(52 141)(53 142)(54 143)(55 144)(56 145)(57 146)(58 147)(59 148)(60 149)(61 150)(62 151)(63 152)(64 153)(65 154)(66 155)(67 156)(68 157)(69 158)(70 159)(71 160)(72 161)(73 162)(74 163)(75 164)(76 165)(77 166)(78 167)(79 168)(80 169)(81 170)(82 171)(83 172)(84 173)(85 174)(86 175)(87 176)(88 133)(177 253)(178 254)(179 255)(180 256)(181 257)(182 258)(183 259)(184 260)(185 261)(186 262)(187 263)(188 264)(189 221)(190 222)(191 223)(192 224)(193 225)(194 226)(195 227)(196 228)(197 229)(198 230)(199 231)(200 232)(201 233)(202 234)(203 235)(204 236)(205 237)(206 238)(207 239)(208 240)(209 241)(210 242)(211 243)(212 244)(213 245)(214 246)(215 247)(216 248)(217 249)(218 250)(219 251)(220 252)(265 326)(266 327)(267 328)(268 329)(269 330)(270 331)(271 332)(272 333)(273 334)(274 335)(275 336)(276 337)(277 338)(278 339)(279 340)(280 341)(281 342)(282 343)(283 344)(284 345)(285 346)(286 347)(287 348)(288 349)(289 350)(290 351)(291 352)(292 309)(293 310)(294 311)(295 312)(296 313)(297 314)(298 315)(299 316)(300 317)(301 318)(302 319)(303 320)(304 321)(305 322)(306 323)(307 324)(308 325)
(1 309 197 82)(2 310 198 83)(3 311 199 84)(4 312 200 85)(5 313 201 86)(6 314 202 87)(7 315 203 88)(8 316 204 45)(9 317 205 46)(10 318 206 47)(11 319 207 48)(12 320 208 49)(13 321 209 50)(14 322 210 51)(15 323 211 52)(16 324 212 53)(17 325 213 54)(18 326 214 55)(19 327 215 56)(20 328 216 57)(21 329 217 58)(22 330 218 59)(23 331 219 60)(24 332 220 61)(25 333 177 62)(26 334 178 63)(27 335 179 64)(28 336 180 65)(29 337 181 66)(30 338 182 67)(31 339 183 68)(32 340 184 69)(33 341 185 70)(34 342 186 71)(35 343 187 72)(36 344 188 73)(37 345 189 74)(38 346 190 75)(39 347 191 76)(40 348 192 77)(41 349 193 78)(42 350 194 79)(43 351 195 80)(44 352 196 81)(89 301 238 136)(90 302 239 137)(91 303 240 138)(92 304 241 139)(93 305 242 140)(94 306 243 141)(95 307 244 142)(96 308 245 143)(97 265 246 144)(98 266 247 145)(99 267 248 146)(100 268 249 147)(101 269 250 148)(102 270 251 149)(103 271 252 150)(104 272 253 151)(105 273 254 152)(106 274 255 153)(107 275 256 154)(108 276 257 155)(109 277 258 156)(110 278 259 157)(111 279 260 158)(112 280 261 159)(113 281 262 160)(114 282 263 161)(115 283 264 162)(116 284 221 163)(117 285 222 164)(118 286 223 165)(119 287 224 166)(120 288 225 167)(121 289 226 168)(122 290 227 169)(123 291 228 170)(124 292 229 171)(125 293 230 172)(126 294 231 173)(127 295 232 174)(128 296 233 175)(129 297 234 176)(130 298 235 133)(131 299 236 134)(132 300 237 135)
(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)| (1,124)(2,125)(3,126)(4,127)(5,128)(6,129)(7,130)(8,131)(9,132)(10,89)(11,90)(12,91)(13,92)(14,93)(15,94)(16,95)(17,96)(18,97)(19,98)(20,99)(21,100)(22,101)(23,102)(24,103)(25,104)(26,105)(27,106)(28,107)(29,108)(30,109)(31,110)(32,111)(33,112)(34,113)(35,114)(36,115)(37,116)(38,117)(39,118)(40,119)(41,120)(42,121)(43,122)(44,123)(45,134)(46,135)(47,136)(48,137)(49,138)(50,139)(51,140)(52,141)(53,142)(54,143)(55,144)(56,145)(57,146)(58,147)(59,148)(60,149)(61,150)(62,151)(63,152)(64,153)(65,154)(66,155)(67,156)(68,157)(69,158)(70,159)(71,160)(72,161)(73,162)(74,163)(75,164)(76,165)(77,166)(78,167)(79,168)(80,169)(81,170)(82,171)(83,172)(84,173)(85,174)(86,175)(87,176)(88,133)(177,253)(178,254)(179,255)(180,256)(181,257)(182,258)(183,259)(184,260)(185,261)(186,262)(187,263)(188,264)(189,221)(190,222)(191,223)(192,224)(193,225)(194,226)(195,227)(196,228)(197,229)(198,230)(199,231)(200,232)(201,233)(202,234)(203,235)(204,236)(205,237)(206,238)(207,239)(208,240)(209,241)(210,242)(211,243)(212,244)(213,245)(214,246)(215,247)(216,248)(217,249)(218,250)(219,251)(220,252)(265,326)(266,327)(267,328)(268,329)(269,330)(270,331)(271,332)(272,333)(273,334)(274,335)(275,336)(276,337)(277,338)(278,339)(279,340)(280,341)(281,342)(282,343)(283,344)(284,345)(285,346)(286,347)(287,348)(288,349)(289,350)(290,351)(291,352)(292,309)(293,310)(294,311)(295,312)(296,313)(297,314)(298,315)(299,316)(300,317)(301,318)(302,319)(303,320)(304,321)(305,322)(306,323)(307,324)(308,325), (1,309,197,82)(2,310,198,83)(3,311,199,84)(4,312,200,85)(5,313,201,86)(6,314,202,87)(7,315,203,88)(8,316,204,45)(9,317,205,46)(10,318,206,47)(11,319,207,48)(12,320,208,49)(13,321,209,50)(14,322,210,51)(15,323,211,52)(16,324,212,53)(17,325,213,54)(18,326,214,55)(19,327,215,56)(20,328,216,57)(21,329,217,58)(22,330,218,59)(23,331,219,60)(24,332,220,61)(25,333,177,62)(26,334,178,63)(27,335,179,64)(28,336,180,65)(29,337,181,66)(30,338,182,67)(31,339,183,68)(32,340,184,69)(33,341,185,70)(34,342,186,71)(35,343,187,72)(36,344,188,73)(37,345,189,74)(38,346,190,75)(39,347,191,76)(40,348,192,77)(41,349,193,78)(42,350,194,79)(43,351,195,80)(44,352,196,81)(89,301,238,136)(90,302,239,137)(91,303,240,138)(92,304,241,139)(93,305,242,140)(94,306,243,141)(95,307,244,142)(96,308,245,143)(97,265,246,144)(98,266,247,145)(99,267,248,146)(100,268,249,147)(101,269,250,148)(102,270,251,149)(103,271,252,150)(104,272,253,151)(105,273,254,152)(106,274,255,153)(107,275,256,154)(108,276,257,155)(109,277,258,156)(110,278,259,157)(111,279,260,158)(112,280,261,159)(113,281,262,160)(114,282,263,161)(115,283,264,162)(116,284,221,163)(117,285,222,164)(118,286,223,165)(119,287,224,166)(120,288,225,167)(121,289,226,168)(122,290,227,169)(123,291,228,170)(124,292,229,171)(125,293,230,172)(126,294,231,173)(127,295,232,174)(128,296,233,175)(129,297,234,176)(130,298,235,133)(131,299,236,134)(132,300,237,135), (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( (1,124)(2,125)(3,126)(4,127)(5,128)(6,129)(7,130)(8,131)(9,132)(10,89)(11,90)(12,91)(13,92)(14,93)(15,94)(16,95)(17,96)(18,97)(19,98)(20,99)(21,100)(22,101)(23,102)(24,103)(25,104)(26,105)(27,106)(28,107)(29,108)(30,109)(31,110)(32,111)(33,112)(34,113)(35,114)(36,115)(37,116)(38,117)(39,118)(40,119)(41,120)(42,121)(43,122)(44,123)(45,134)(46,135)(47,136)(48,137)(49,138)(50,139)(51,140)(52,141)(53,142)(54,143)(55,144)(56,145)(57,146)(58,147)(59,148)(60,149)(61,150)(62,151)(63,152)(64,153)(65,154)(66,155)(67,156)(68,157)(69,158)(70,159)(71,160)(72,161)(73,162)(74,163)(75,164)(76,165)(77,166)(78,167)(79,168)(80,169)(81,170)(82,171)(83,172)(84,173)(85,174)(86,175)(87,176)(88,133)(177,253)(178,254)(179,255)(180,256)(181,257)(182,258)(183,259)(184,260)(185,261)(186,262)(187,263)(188,264)(189,221)(190,222)(191,223)(192,224)(193,225)(194,226)(195,227)(196,228)(197,229)(198,230)(199,231)(200,232)(201,233)(202,234)(203,235)(204,236)(205,237)(206,238)(207,239)(208,240)(209,241)(210,242)(211,243)(212,244)(213,245)(214,246)(215,247)(216,248)(217,249)(218,250)(219,251)(220,252)(265,326)(266,327)(267,328)(268,329)(269,330)(270,331)(271,332)(272,333)(273,334)(274,335)(275,336)(276,337)(277,338)(278,339)(279,340)(280,341)(281,342)(282,343)(283,344)(284,345)(285,346)(286,347)(287,348)(288,349)(289,350)(290,351)(291,352)(292,309)(293,310)(294,311)(295,312)(296,313)(297,314)(298,315)(299,316)(300,317)(301,318)(302,319)(303,320)(304,321)(305,322)(306,323)(307,324)(308,325), (1,309,197,82)(2,310,198,83)(3,311,199,84)(4,312,200,85)(5,313,201,86)(6,314,202,87)(7,315,203,88)(8,316,204,45)(9,317,205,46)(10,318,206,47)(11,319,207,48)(12,320,208,49)(13,321,209,50)(14,322,210,51)(15,323,211,52)(16,324,212,53)(17,325,213,54)(18,326,214,55)(19,327,215,56)(20,328,216,57)(21,329,217,58)(22,330,218,59)(23,331,219,60)(24,332,220,61)(25,333,177,62)(26,334,178,63)(27,335,179,64)(28,336,180,65)(29,337,181,66)(30,338,182,67)(31,339,183,68)(32,340,184,69)(33,341,185,70)(34,342,186,71)(35,343,187,72)(36,344,188,73)(37,345,189,74)(38,346,190,75)(39,347,191,76)(40,348,192,77)(41,349,193,78)(42,350,194,79)(43,351,195,80)(44,352,196,81)(89,301,238,136)(90,302,239,137)(91,303,240,138)(92,304,241,139)(93,305,242,140)(94,306,243,141)(95,307,244,142)(96,308,245,143)(97,265,246,144)(98,266,247,145)(99,267,248,146)(100,268,249,147)(101,269,250,148)(102,270,251,149)(103,271,252,150)(104,272,253,151)(105,273,254,152)(106,274,255,153)(107,275,256,154)(108,276,257,155)(109,277,258,156)(110,278,259,157)(111,279,260,158)(112,280,261,159)(113,281,262,160)(114,282,263,161)(115,283,264,162)(116,284,221,163)(117,285,222,164)(118,286,223,165)(119,287,224,166)(120,288,225,167)(121,289,226,168)(122,290,227,169)(123,291,228,170)(124,292,229,171)(125,293,230,172)(126,294,231,173)(127,295,232,174)(128,296,233,175)(129,297,234,176)(130,298,235,133)(131,299,236,134)(132,300,237,135), (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([[(1,124),(2,125),(3,126),(4,127),(5,128),(6,129),(7,130),(8,131),(9,132),(10,89),(11,90),(12,91),(13,92),(14,93),(15,94),(16,95),(17,96),(18,97),(19,98),(20,99),(21,100),(22,101),(23,102),(24,103),(25,104),(26,105),(27,106),(28,107),(29,108),(30,109),(31,110),(32,111),(33,112),(34,113),(35,114),(36,115),(37,116),(38,117),(39,118),(40,119),(41,120),(42,121),(43,122),(44,123),(45,134),(46,135),(47,136),(48,137),(49,138),(50,139),(51,140),(52,141),(53,142),(54,143),(55,144),(56,145),(57,146),(58,147),(59,148),(60,149),(61,150),(62,151),(63,152),(64,153),(65,154),(66,155),(67,156),(68,157),(69,158),(70,159),(71,160),(72,161),(73,162),(74,163),(75,164),(76,165),(77,166),(78,167),(79,168),(80,169),(81,170),(82,171),(83,172),(84,173),(85,174),(86,175),(87,176),(88,133),(177,253),(178,254),(179,255),(180,256),(181,257),(182,258),(183,259),(184,260),(185,261),(186,262),(187,263),(188,264),(189,221),(190,222),(191,223),(192,224),(193,225),(194,226),(195,227),(196,228),(197,229),(198,230),(199,231),(200,232),(201,233),(202,234),(203,235),(204,236),(205,237),(206,238),(207,239),(208,240),(209,241),(210,242),(211,243),(212,244),(213,245),(214,246),(215,247),(216,248),(217,249),(218,250),(219,251),(220,252),(265,326),(266,327),(267,328),(268,329),(269,330),(270,331),(271,332),(272,333),(273,334),(274,335),(275,336),(276,337),(277,338),(278,339),(279,340),(280,341),(281,342),(282,343),(283,344),(284,345),(285,346),(286,347),(287,348),(288,349),(289,350),(290,351),(291,352),(292,309),(293,310),(294,311),(295,312),(296,313),(297,314),(298,315),(299,316),(300,317),(301,318),(302,319),(303,320),(304,321),(305,322),(306,323),(307,324),(308,325)], [(1,309,197,82),(2,310,198,83),(3,311,199,84),(4,312,200,85),(5,313,201,86),(6,314,202,87),(7,315,203,88),(8,316,204,45),(9,317,205,46),(10,318,206,47),(11,319,207,48),(12,320,208,49),(13,321,209,50),(14,322,210,51),(15,323,211,52),(16,324,212,53),(17,325,213,54),(18,326,214,55),(19,327,215,56),(20,328,216,57),(21,329,217,58),(22,330,218,59),(23,331,219,60),(24,332,220,61),(25,333,177,62),(26,334,178,63),(27,335,179,64),(28,336,180,65),(29,337,181,66),(30,338,182,67),(31,339,183,68),(32,340,184,69),(33,341,185,70),(34,342,186,71),(35,343,187,72),(36,344,188,73),(37,345,189,74),(38,346,190,75),(39,347,191,76),(40,348,192,77),(41,349,193,78),(42,350,194,79),(43,351,195,80),(44,352,196,81),(89,301,238,136),(90,302,239,137),(91,303,240,138),(92,304,241,139),(93,305,242,140),(94,306,243,141),(95,307,244,142),(96,308,245,143),(97,265,246,144),(98,266,247,145),(99,267,248,146),(100,268,249,147),(101,269,250,148),(102,270,251,149),(103,271,252,150),(104,272,253,151),(105,273,254,152),(106,274,255,153),(107,275,256,154),(108,276,257,155),(109,277,258,156),(110,278,259,157),(111,279,260,158),(112,280,261,159),(113,281,262,160),(114,282,263,161),(115,283,264,162),(116,284,221,163),(117,285,222,164),(118,286,223,165),(119,287,224,166),(120,288,225,167),(121,289,226,168),(122,290,227,169),(123,291,228,170),(124,292,229,171),(125,293,230,172),(126,294,231,173),(127,295,232,174),(128,296,233,175),(129,297,234,176),(130,298,235,133),(131,299,236,134),(132,300,237,135)], [(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 ··· 2G 4A ··· 4X 11A ··· 11J 22A ··· 22BR 44A ··· 44IF order 1 2 ··· 2 4 ··· 4 11 ··· 11 22 ··· 22 44 ··· 44 size 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

352 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C4 C11 C22 C22 C44 kernel C2×C4×C44 C4×C44 C22×C44 C2×C44 C2×C42 C42 C22×C4 C2×C4 # reps 1 4 3 24 10 40 30 240

Matrix representation of C2×C4×C44 in GL3(𝔽89) generated by

 1 0 0 0 1 0 0 0 88
,
 55 0 0 0 55 0 0 0 55
,
 44 0 0 0 18 0 0 0 80
G:=sub<GL(3,GF(89))| [1,0,0,0,1,0,0,0,88],[55,0,0,0,55,0,0,0,55],[44,0,0,0,18,0,0,0,80] >;

C2×C4×C44 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_{44}
% in TeX

G:=Group("C2xC4xC44");
// GroupNames label

G:=SmallGroup(352,149);
// by ID

G=gap.SmallGroup(352,149);
# by ID

G:=PCGroup([6,-2,-2,-2,-11,-2,-2,528,1063]);
// Polycyclic

G:=Group<a,b,c|a^2=b^4=c^44=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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