metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C44.3Q8, C4.3Dic22, C4⋊C4.6D11, C22.6(C2×Q8), (C2×C4).43D22, C44⋊C4.7C2, C2.8(C2×Dic22), C11⋊3(C42.C2), C22.25(C4○D4), Dic11⋊C4.3C2, (C2×C22).31C23, (C2×C44).22C22, (C4×Dic11).2C2, C2.4(D44⋊C2), C2.12(D4⋊2D11), C22.48(C22×D11), (C2×Dic11).10C22, (C11×C4⋊C4).7C2, SmallGroup(352,85)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C44.3Q8
G = < a,b,c | a4=b44=1, c2=b22, bab-1=a-1, ac=ca, cbc-1=a2b-1 >
Subgroups: 258 in 56 conjugacy classes, 33 normal (19 characteristic)
C1, C2, C4, C4, C22, C2×C4, C2×C4, C2×C4, C11, C42, C4⋊C4, C4⋊C4, C22, C42.C2, Dic11, C44, C44, C2×C22, C2×Dic11, C2×Dic11, C2×C44, C2×C44, C4×Dic11, Dic11⋊C4, C44⋊C4, C44⋊C4, C11×C4⋊C4, C44.3Q8
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, D11, C42.C2, D22, Dic22, C22×D11, C2×Dic22, D4⋊2D11, D44⋊C2, C44.3Q8
►Regular action on 352 points
Generators in S352
(1 169 261 330)(2 331 262 170)(3 171 263 332)(4 333 264 172)(5 173 221 334)(6 335 222 174)(7 175 223 336)(8 337 224 176)(9 133 225 338)(10 339 226 134)(11 135 227 340)(12 341 228 136)(13 137 229 342)(14 343 230 138)(15 139 231 344)(16 345 232 140)(17 141 233 346)(18 347 234 142)(19 143 235 348)(20 349 236 144)(21 145 237 350)(22 351 238 146)(23 147 239 352)(24 309 240 148)(25 149 241 310)(26 311 242 150)(27 151 243 312)(28 313 244 152)(29 153 245 314)(30 315 246 154)(31 155 247 316)(32 317 248 156)(33 157 249 318)(34 319 250 158)(35 159 251 320)(36 321 252 160)(37 161 253 322)(38 323 254 162)(39 163 255 324)(40 325 256 164)(41 165 257 326)(42 327 258 166)(43 167 259 328)(44 329 260 168)(45 284 180 91)(46 92 181 285)(47 286 182 93)(48 94 183 287)(49 288 184 95)(50 96 185 289)(51 290 186 97)(52 98 187 291)(53 292 188 99)(54 100 189 293)(55 294 190 101)(56 102 191 295)(57 296 192 103)(58 104 193 297)(59 298 194 105)(60 106 195 299)(61 300 196 107)(62 108 197 301)(63 302 198 109)(64 110 199 303)(65 304 200 111)(66 112 201 305)(67 306 202 113)(68 114 203 307)(69 308 204 115)(70 116 205 265)(71 266 206 117)(72 118 207 267)(73 268 208 119)(74 120 209 269)(75 270 210 121)(76 122 211 271)(77 272 212 123)(78 124 213 273)(79 274 214 125)(80 126 215 275)(81 276 216 127)(82 128 217 277)(83 278 218 129)(84 130 219 279)(85 280 220 131)(86 132 177 281)(87 282 178 89)(88 90 179 283)
(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 84 23 62)(2 218 24 196)(3 82 25 60)(4 216 26 194)(5 80 27 58)(6 214 28 192)(7 78 29 56)(8 212 30 190)(9 76 31 54)(10 210 32 188)(11 74 33 52)(12 208 34 186)(13 72 35 50)(14 206 36 184)(15 70 37 48)(16 204 38 182)(17 68 39 46)(18 202 40 180)(19 66 41 88)(20 200 42 178)(21 64 43 86)(22 198 44 220)(45 234 67 256)(47 232 69 254)(49 230 71 252)(51 228 73 250)(53 226 75 248)(55 224 77 246)(57 222 79 244)(59 264 81 242)(61 262 83 240)(63 260 85 238)(65 258 87 236)(89 349 111 327)(90 143 112 165)(91 347 113 325)(92 141 114 163)(93 345 115 323)(94 139 116 161)(95 343 117 321)(96 137 118 159)(97 341 119 319)(98 135 120 157)(99 339 121 317)(100 133 122 155)(101 337 123 315)(102 175 124 153)(103 335 125 313)(104 173 126 151)(105 333 127 311)(106 171 128 149)(107 331 129 309)(108 169 130 147)(109 329 131 351)(110 167 132 145)(134 270 156 292)(136 268 158 290)(138 266 160 288)(140 308 162 286)(142 306 164 284)(144 304 166 282)(146 302 168 280)(148 300 170 278)(150 298 172 276)(152 296 174 274)(154 294 176 272)(177 237 199 259)(179 235 201 257)(181 233 203 255)(183 231 205 253)(185 229 207 251)(187 227 209 249)(189 225 211 247)(191 223 213 245)(193 221 215 243)(195 263 217 241)(197 261 219 239)(265 322 287 344)(267 320 289 342)(269 318 291 340)(271 316 293 338)(273 314 295 336)(275 312 297 334)(277 310 299 332)(279 352 301 330)(281 350 303 328)(283 348 305 326)(285 346 307 324)
G:=sub<Sym(352)| (1,169,261,330)(2,331,262,170)(3,171,263,332)(4,333,264,172)(5,173,221,334)(6,335,222,174)(7,175,223,336)(8,337,224,176)(9,133,225,338)(10,339,226,134)(11,135,227,340)(12,341,228,136)(13,137,229,342)(14,343,230,138)(15,139,231,344)(16,345,232,140)(17,141,233,346)(18,347,234,142)(19,143,235,348)(20,349,236,144)(21,145,237,350)(22,351,238,146)(23,147,239,352)(24,309,240,148)(25,149,241,310)(26,311,242,150)(27,151,243,312)(28,313,244,152)(29,153,245,314)(30,315,246,154)(31,155,247,316)(32,317,248,156)(33,157,249,318)(34,319,250,158)(35,159,251,320)(36,321,252,160)(37,161,253,322)(38,323,254,162)(39,163,255,324)(40,325,256,164)(41,165,257,326)(42,327,258,166)(43,167,259,328)(44,329,260,168)(45,284,180,91)(46,92,181,285)(47,286,182,93)(48,94,183,287)(49,288,184,95)(50,96,185,289)(51,290,186,97)(52,98,187,291)(53,292,188,99)(54,100,189,293)(55,294,190,101)(56,102,191,295)(57,296,192,103)(58,104,193,297)(59,298,194,105)(60,106,195,299)(61,300,196,107)(62,108,197,301)(63,302,198,109)(64,110,199,303)(65,304,200,111)(66,112,201,305)(67,306,202,113)(68,114,203,307)(69,308,204,115)(70,116,205,265)(71,266,206,117)(72,118,207,267)(73,268,208,119)(74,120,209,269)(75,270,210,121)(76,122,211,271)(77,272,212,123)(78,124,213,273)(79,274,214,125)(80,126,215,275)(81,276,216,127)(82,128,217,277)(83,278,218,129)(84,130,219,279)(85,280,220,131)(86,132,177,281)(87,282,178,89)(88,90,179,283), (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), (1,84,23,62)(2,218,24,196)(3,82,25,60)(4,216,26,194)(5,80,27,58)(6,214,28,192)(7,78,29,56)(8,212,30,190)(9,76,31,54)(10,210,32,188)(11,74,33,52)(12,208,34,186)(13,72,35,50)(14,206,36,184)(15,70,37,48)(16,204,38,182)(17,68,39,46)(18,202,40,180)(19,66,41,88)(20,200,42,178)(21,64,43,86)(22,198,44,220)(45,234,67,256)(47,232,69,254)(49,230,71,252)(51,228,73,250)(53,226,75,248)(55,224,77,246)(57,222,79,244)(59,264,81,242)(61,262,83,240)(63,260,85,238)(65,258,87,236)(89,349,111,327)(90,143,112,165)(91,347,113,325)(92,141,114,163)(93,345,115,323)(94,139,116,161)(95,343,117,321)(96,137,118,159)(97,341,119,319)(98,135,120,157)(99,339,121,317)(100,133,122,155)(101,337,123,315)(102,175,124,153)(103,335,125,313)(104,173,126,151)(105,333,127,311)(106,171,128,149)(107,331,129,309)(108,169,130,147)(109,329,131,351)(110,167,132,145)(134,270,156,292)(136,268,158,290)(138,266,160,288)(140,308,162,286)(142,306,164,284)(144,304,166,282)(146,302,168,280)(148,300,170,278)(150,298,172,276)(152,296,174,274)(154,294,176,272)(177,237,199,259)(179,235,201,257)(181,233,203,255)(183,231,205,253)(185,229,207,251)(187,227,209,249)(189,225,211,247)(191,223,213,245)(193,221,215,243)(195,263,217,241)(197,261,219,239)(265,322,287,344)(267,320,289,342)(269,318,291,340)(271,316,293,338)(273,314,295,336)(275,312,297,334)(277,310,299,332)(279,352,301,330)(281,350,303,328)(283,348,305,326)(285,346,307,324)>;
G:=Group( (1,169,261,330)(2,331,262,170)(3,171,263,332)(4,333,264,172)(5,173,221,334)(6,335,222,174)(7,175,223,336)(8,337,224,176)(9,133,225,338)(10,339,226,134)(11,135,227,340)(12,341,228,136)(13,137,229,342)(14,343,230,138)(15,139,231,344)(16,345,232,140)(17,141,233,346)(18,347,234,142)(19,143,235,348)(20,349,236,144)(21,145,237,350)(22,351,238,146)(23,147,239,352)(24,309,240,148)(25,149,241,310)(26,311,242,150)(27,151,243,312)(28,313,244,152)(29,153,245,314)(30,315,246,154)(31,155,247,316)(32,317,248,156)(33,157,249,318)(34,319,250,158)(35,159,251,320)(36,321,252,160)(37,161,253,322)(38,323,254,162)(39,163,255,324)(40,325,256,164)(41,165,257,326)(42,327,258,166)(43,167,259,328)(44,329,260,168)(45,284,180,91)(46,92,181,285)(47,286,182,93)(48,94,183,287)(49,288,184,95)(50,96,185,289)(51,290,186,97)(52,98,187,291)(53,292,188,99)(54,100,189,293)(55,294,190,101)(56,102,191,295)(57,296,192,103)(58,104,193,297)(59,298,194,105)(60,106,195,299)(61,300,196,107)(62,108,197,301)(63,302,198,109)(64,110,199,303)(65,304,200,111)(66,112,201,305)(67,306,202,113)(68,114,203,307)(69,308,204,115)(70,116,205,265)(71,266,206,117)(72,118,207,267)(73,268,208,119)(74,120,209,269)(75,270,210,121)(76,122,211,271)(77,272,212,123)(78,124,213,273)(79,274,214,125)(80,126,215,275)(81,276,216,127)(82,128,217,277)(83,278,218,129)(84,130,219,279)(85,280,220,131)(86,132,177,281)(87,282,178,89)(88,90,179,283), (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), (1,84,23,62)(2,218,24,196)(3,82,25,60)(4,216,26,194)(5,80,27,58)(6,214,28,192)(7,78,29,56)(8,212,30,190)(9,76,31,54)(10,210,32,188)(11,74,33,52)(12,208,34,186)(13,72,35,50)(14,206,36,184)(15,70,37,48)(16,204,38,182)(17,68,39,46)(18,202,40,180)(19,66,41,88)(20,200,42,178)(21,64,43,86)(22,198,44,220)(45,234,67,256)(47,232,69,254)(49,230,71,252)(51,228,73,250)(53,226,75,248)(55,224,77,246)(57,222,79,244)(59,264,81,242)(61,262,83,240)(63,260,85,238)(65,258,87,236)(89,349,111,327)(90,143,112,165)(91,347,113,325)(92,141,114,163)(93,345,115,323)(94,139,116,161)(95,343,117,321)(96,137,118,159)(97,341,119,319)(98,135,120,157)(99,339,121,317)(100,133,122,155)(101,337,123,315)(102,175,124,153)(103,335,125,313)(104,173,126,151)(105,333,127,311)(106,171,128,149)(107,331,129,309)(108,169,130,147)(109,329,131,351)(110,167,132,145)(134,270,156,292)(136,268,158,290)(138,266,160,288)(140,308,162,286)(142,306,164,284)(144,304,166,282)(146,302,168,280)(148,300,170,278)(150,298,172,276)(152,296,174,274)(154,294,176,272)(177,237,199,259)(179,235,201,257)(181,233,203,255)(183,231,205,253)(185,229,207,251)(187,227,209,249)(189,225,211,247)(191,223,213,245)(193,221,215,243)(195,263,217,241)(197,261,219,239)(265,322,287,344)(267,320,289,342)(269,318,291,340)(271,316,293,338)(273,314,295,336)(275,312,297,334)(277,310,299,332)(279,352,301,330)(281,350,303,328)(283,348,305,326)(285,346,307,324) );
G=PermutationGroup([[(1,169,261,330),(2,331,262,170),(3,171,263,332),(4,333,264,172),(5,173,221,334),(6,335,222,174),(7,175,223,336),(8,337,224,176),(9,133,225,338),(10,339,226,134),(11,135,227,340),(12,341,228,136),(13,137,229,342),(14,343,230,138),(15,139,231,344),(16,345,232,140),(17,141,233,346),(18,347,234,142),(19,143,235,348),(20,349,236,144),(21,145,237,350),(22,351,238,146),(23,147,239,352),(24,309,240,148),(25,149,241,310),(26,311,242,150),(27,151,243,312),(28,313,244,152),(29,153,245,314),(30,315,246,154),(31,155,247,316),(32,317,248,156),(33,157,249,318),(34,319,250,158),(35,159,251,320),(36,321,252,160),(37,161,253,322),(38,323,254,162),(39,163,255,324),(40,325,256,164),(41,165,257,326),(42,327,258,166),(43,167,259,328),(44,329,260,168),(45,284,180,91),(46,92,181,285),(47,286,182,93),(48,94,183,287),(49,288,184,95),(50,96,185,289),(51,290,186,97),(52,98,187,291),(53,292,188,99),(54,100,189,293),(55,294,190,101),(56,102,191,295),(57,296,192,103),(58,104,193,297),(59,298,194,105),(60,106,195,299),(61,300,196,107),(62,108,197,301),(63,302,198,109),(64,110,199,303),(65,304,200,111),(66,112,201,305),(67,306,202,113),(68,114,203,307),(69,308,204,115),(70,116,205,265),(71,266,206,117),(72,118,207,267),(73,268,208,119),(74,120,209,269),(75,270,210,121),(76,122,211,271),(77,272,212,123),(78,124,213,273),(79,274,214,125),(80,126,215,275),(81,276,216,127),(82,128,217,277),(83,278,218,129),(84,130,219,279),(85,280,220,131),(86,132,177,281),(87,282,178,89),(88,90,179,283)], [(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)], [(1,84,23,62),(2,218,24,196),(3,82,25,60),(4,216,26,194),(5,80,27,58),(6,214,28,192),(7,78,29,56),(8,212,30,190),(9,76,31,54),(10,210,32,188),(11,74,33,52),(12,208,34,186),(13,72,35,50),(14,206,36,184),(15,70,37,48),(16,204,38,182),(17,68,39,46),(18,202,40,180),(19,66,41,88),(20,200,42,178),(21,64,43,86),(22,198,44,220),(45,234,67,256),(47,232,69,254),(49,230,71,252),(51,228,73,250),(53,226,75,248),(55,224,77,246),(57,222,79,244),(59,264,81,242),(61,262,83,240),(63,260,85,238),(65,258,87,236),(89,349,111,327),(90,143,112,165),(91,347,113,325),(92,141,114,163),(93,345,115,323),(94,139,116,161),(95,343,117,321),(96,137,118,159),(97,341,119,319),(98,135,120,157),(99,339,121,317),(100,133,122,155),(101,337,123,315),(102,175,124,153),(103,335,125,313),(104,173,126,151),(105,333,127,311),(106,171,128,149),(107,331,129,309),(108,169,130,147),(109,329,131,351),(110,167,132,145),(134,270,156,292),(136,268,158,290),(138,266,160,288),(140,308,162,286),(142,306,164,284),(144,304,166,282),(146,302,168,280),(148,300,170,278),(150,298,172,276),(152,296,174,274),(154,294,176,272),(177,237,199,259),(179,235,201,257),(181,233,203,255),(183,231,205,253),(185,229,207,251),(187,227,209,249),(189,225,211,247),(191,223,213,245),(193,221,215,243),(195,263,217,241),(197,261,219,239),(265,322,287,344),(267,320,289,342),(269,318,291,340),(271,316,293,338),(273,314,295,336),(275,312,297,334),(277,310,299,332),(279,352,301,330),(281,350,303,328),(283,348,305,326),(285,346,307,324)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44AD |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 22 | 22 | 22 | 22 | 44 | 44 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | Q8 | C4○D4 | D11 | D22 | Dic22 | D4⋊2D11 | D44⋊C2 |
| kernel | C44.3Q8 | C4×Dic11 | Dic11⋊C4 | C44⋊C4 | C11×C4⋊C4 | C44 | C22 | C4⋊C4 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 2 | 3 | 1 | 2 | 4 | 5 | 15 | 20 | 5 | 5 |
Matrix representation of C44.3Q8 ►in GL6(𝔽89)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 48 | 2 | 0 | 0 |
| 0 | 0 | 49 | 41 | 0 | 0 |
| 0 | 0 | 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 0 | 0 | 88 |
| 1 | 88 | 0 | 0 | 0 | 0 |
| 2 | 88 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 87 | 0 | 0 |
| 0 | 0 | 1 | 88 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 88 |
| 0 | 0 | 0 | 0 | 73 | 17 |
| 14 | 59 | 0 | 0 | 0 | 0 |
| 57 | 75 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 0 | 0 | 0 |
| 0 | 0 | 0 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 |
G:=sub<GL(6,GF(89))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,48,49,0,0,0,0,2,41,0,0,0,0,0,0,88,0,0,0,0,0,0,88],[1,2,0,0,0,0,88,88,0,0,0,0,0,0,1,1,0,0,0,0,87,88,0,0,0,0,0,0,1,73,0,0,0,0,88,17],[14,57,0,0,0,0,59,75,0,0,0,0,0,0,34,0,0,0,0,0,0,34,0,0,0,0,0,0,88,16,0,0,0,0,0,1] >;
C44.3Q8 in GAP, Magma, Sage, TeX
C_{44}._3Q_8 % in TeX
G:=Group("C44.3Q8"); // GroupNames label
G:=SmallGroup(352,85);
// by ID
G=gap.SmallGroup(352,85);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,48,103,506,188,50,11525]);
// Polycyclic
G:=Group<a,b,c|a^4=b^44=1,c^2=b^22,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b^-1>;
// generators/relations