direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C45, C4.C90, C36.3C10, C180.7C2, C60.12C6, C12.4C30, C20.3C18, C90.24C22, C3.(Q8×C15), C2.2(C2×C90), C6.7(C2×C30), C15.2(C3×Q8), C30.30(C2×C6), C10.7(C2×C18), C18.7(C2×C10), (Q8×C15).2C3, (C3×Q8).2C15, SmallGroup(360,32)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C45
G = < a,b,c | a45=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C45
►Regular action on 360 points
Generators in S360
(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)(316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360)
(1 284 256 204)(2 285 257 205)(3 286 258 206)(4 287 259 207)(5 288 260 208)(6 289 261 209)(7 290 262 210)(8 291 263 211)(9 292 264 212)(10 293 265 213)(11 294 266 214)(12 295 267 215)(13 296 268 216)(14 297 269 217)(15 298 270 218)(16 299 226 219)(17 300 227 220)(18 301 228 221)(19 302 229 222)(20 303 230 223)(21 304 231 224)(22 305 232 225)(23 306 233 181)(24 307 234 182)(25 308 235 183)(26 309 236 184)(27 310 237 185)(28 311 238 186)(29 312 239 187)(30 313 240 188)(31 314 241 189)(32 315 242 190)(33 271 243 191)(34 272 244 192)(35 273 245 193)(36 274 246 194)(37 275 247 195)(38 276 248 196)(39 277 249 197)(40 278 250 198)(41 279 251 199)(42 280 252 200)(43 281 253 201)(44 282 254 202)(45 283 255 203)(46 357 147 103)(47 358 148 104)(48 359 149 105)(49 360 150 106)(50 316 151 107)(51 317 152 108)(52 318 153 109)(53 319 154 110)(54 320 155 111)(55 321 156 112)(56 322 157 113)(57 323 158 114)(58 324 159 115)(59 325 160 116)(60 326 161 117)(61 327 162 118)(62 328 163 119)(63 329 164 120)(64 330 165 121)(65 331 166 122)(66 332 167 123)(67 333 168 124)(68 334 169 125)(69 335 170 126)(70 336 171 127)(71 337 172 128)(72 338 173 129)(73 339 174 130)(74 340 175 131)(75 341 176 132)(76 342 177 133)(77 343 178 134)(78 344 179 135)(79 345 180 91)(80 346 136 92)(81 347 137 93)(82 348 138 94)(83 349 139 95)(84 350 140 96)(85 351 141 97)(86 352 142 98)(87 353 143 99)(88 354 144 100)(89 355 145 101)(90 356 146 102)
(1 47 256 148)(2 48 257 149)(3 49 258 150)(4 50 259 151)(5 51 260 152)(6 52 261 153)(7 53 262 154)(8 54 263 155)(9 55 264 156)(10 56 265 157)(11 57 266 158)(12 58 267 159)(13 59 268 160)(14 60 269 161)(15 61 270 162)(16 62 226 163)(17 63 227 164)(18 64 228 165)(19 65 229 166)(20 66 230 167)(21 67 231 168)(22 68 232 169)(23 69 233 170)(24 70 234 171)(25 71 235 172)(26 72 236 173)(27 73 237 174)(28 74 238 175)(29 75 239 176)(30 76 240 177)(31 77 241 178)(32 78 242 179)(33 79 243 180)(34 80 244 136)(35 81 245 137)(36 82 246 138)(37 83 247 139)(38 84 248 140)(39 85 249 141)(40 86 250 142)(41 87 251 143)(42 88 252 144)(43 89 253 145)(44 90 254 146)(45 46 255 147)(91 191 345 271)(92 192 346 272)(93 193 347 273)(94 194 348 274)(95 195 349 275)(96 196 350 276)(97 197 351 277)(98 198 352 278)(99 199 353 279)(100 200 354 280)(101 201 355 281)(102 202 356 282)(103 203 357 283)(104 204 358 284)(105 205 359 285)(106 206 360 286)(107 207 316 287)(108 208 317 288)(109 209 318 289)(110 210 319 290)(111 211 320 291)(112 212 321 292)(113 213 322 293)(114 214 323 294)(115 215 324 295)(116 216 325 296)(117 217 326 297)(118 218 327 298)(119 219 328 299)(120 220 329 300)(121 221 330 301)(122 222 331 302)(123 223 332 303)(124 224 333 304)(125 225 334 305)(126 181 335 306)(127 182 336 307)(128 183 337 308)(129 184 338 309)(130 185 339 310)(131 186 340 311)(132 187 341 312)(133 188 342 313)(134 189 343 314)(135 190 344 315)
G:=sub<Sym(360)| (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)(316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360), (1,284,256,204)(2,285,257,205)(3,286,258,206)(4,287,259,207)(5,288,260,208)(6,289,261,209)(7,290,262,210)(8,291,263,211)(9,292,264,212)(10,293,265,213)(11,294,266,214)(12,295,267,215)(13,296,268,216)(14,297,269,217)(15,298,270,218)(16,299,226,219)(17,300,227,220)(18,301,228,221)(19,302,229,222)(20,303,230,223)(21,304,231,224)(22,305,232,225)(23,306,233,181)(24,307,234,182)(25,308,235,183)(26,309,236,184)(27,310,237,185)(28,311,238,186)(29,312,239,187)(30,313,240,188)(31,314,241,189)(32,315,242,190)(33,271,243,191)(34,272,244,192)(35,273,245,193)(36,274,246,194)(37,275,247,195)(38,276,248,196)(39,277,249,197)(40,278,250,198)(41,279,251,199)(42,280,252,200)(43,281,253,201)(44,282,254,202)(45,283,255,203)(46,357,147,103)(47,358,148,104)(48,359,149,105)(49,360,150,106)(50,316,151,107)(51,317,152,108)(52,318,153,109)(53,319,154,110)(54,320,155,111)(55,321,156,112)(56,322,157,113)(57,323,158,114)(58,324,159,115)(59,325,160,116)(60,326,161,117)(61,327,162,118)(62,328,163,119)(63,329,164,120)(64,330,165,121)(65,331,166,122)(66,332,167,123)(67,333,168,124)(68,334,169,125)(69,335,170,126)(70,336,171,127)(71,337,172,128)(72,338,173,129)(73,339,174,130)(74,340,175,131)(75,341,176,132)(76,342,177,133)(77,343,178,134)(78,344,179,135)(79,345,180,91)(80,346,136,92)(81,347,137,93)(82,348,138,94)(83,349,139,95)(84,350,140,96)(85,351,141,97)(86,352,142,98)(87,353,143,99)(88,354,144,100)(89,355,145,101)(90,356,146,102), (1,47,256,148)(2,48,257,149)(3,49,258,150)(4,50,259,151)(5,51,260,152)(6,52,261,153)(7,53,262,154)(8,54,263,155)(9,55,264,156)(10,56,265,157)(11,57,266,158)(12,58,267,159)(13,59,268,160)(14,60,269,161)(15,61,270,162)(16,62,226,163)(17,63,227,164)(18,64,228,165)(19,65,229,166)(20,66,230,167)(21,67,231,168)(22,68,232,169)(23,69,233,170)(24,70,234,171)(25,71,235,172)(26,72,236,173)(27,73,237,174)(28,74,238,175)(29,75,239,176)(30,76,240,177)(31,77,241,178)(32,78,242,179)(33,79,243,180)(34,80,244,136)(35,81,245,137)(36,82,246,138)(37,83,247,139)(38,84,248,140)(39,85,249,141)(40,86,250,142)(41,87,251,143)(42,88,252,144)(43,89,253,145)(44,90,254,146)(45,46,255,147)(91,191,345,271)(92,192,346,272)(93,193,347,273)(94,194,348,274)(95,195,349,275)(96,196,350,276)(97,197,351,277)(98,198,352,278)(99,199,353,279)(100,200,354,280)(101,201,355,281)(102,202,356,282)(103,203,357,283)(104,204,358,284)(105,205,359,285)(106,206,360,286)(107,207,316,287)(108,208,317,288)(109,209,318,289)(110,210,319,290)(111,211,320,291)(112,212,321,292)(113,213,322,293)(114,214,323,294)(115,215,324,295)(116,216,325,296)(117,217,326,297)(118,218,327,298)(119,219,328,299)(120,220,329,300)(121,221,330,301)(122,222,331,302)(123,223,332,303)(124,224,333,304)(125,225,334,305)(126,181,335,306)(127,182,336,307)(128,183,337,308)(129,184,338,309)(130,185,339,310)(131,186,340,311)(132,187,341,312)(133,188,342,313)(134,189,343,314)(135,190,344,315)>;
G:=Group( (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)(316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360), (1,284,256,204)(2,285,257,205)(3,286,258,206)(4,287,259,207)(5,288,260,208)(6,289,261,209)(7,290,262,210)(8,291,263,211)(9,292,264,212)(10,293,265,213)(11,294,266,214)(12,295,267,215)(13,296,268,216)(14,297,269,217)(15,298,270,218)(16,299,226,219)(17,300,227,220)(18,301,228,221)(19,302,229,222)(20,303,230,223)(21,304,231,224)(22,305,232,225)(23,306,233,181)(24,307,234,182)(25,308,235,183)(26,309,236,184)(27,310,237,185)(28,311,238,186)(29,312,239,187)(30,313,240,188)(31,314,241,189)(32,315,242,190)(33,271,243,191)(34,272,244,192)(35,273,245,193)(36,274,246,194)(37,275,247,195)(38,276,248,196)(39,277,249,197)(40,278,250,198)(41,279,251,199)(42,280,252,200)(43,281,253,201)(44,282,254,202)(45,283,255,203)(46,357,147,103)(47,358,148,104)(48,359,149,105)(49,360,150,106)(50,316,151,107)(51,317,152,108)(52,318,153,109)(53,319,154,110)(54,320,155,111)(55,321,156,112)(56,322,157,113)(57,323,158,114)(58,324,159,115)(59,325,160,116)(60,326,161,117)(61,327,162,118)(62,328,163,119)(63,329,164,120)(64,330,165,121)(65,331,166,122)(66,332,167,123)(67,333,168,124)(68,334,169,125)(69,335,170,126)(70,336,171,127)(71,337,172,128)(72,338,173,129)(73,339,174,130)(74,340,175,131)(75,341,176,132)(76,342,177,133)(77,343,178,134)(78,344,179,135)(79,345,180,91)(80,346,136,92)(81,347,137,93)(82,348,138,94)(83,349,139,95)(84,350,140,96)(85,351,141,97)(86,352,142,98)(87,353,143,99)(88,354,144,100)(89,355,145,101)(90,356,146,102), (1,47,256,148)(2,48,257,149)(3,49,258,150)(4,50,259,151)(5,51,260,152)(6,52,261,153)(7,53,262,154)(8,54,263,155)(9,55,264,156)(10,56,265,157)(11,57,266,158)(12,58,267,159)(13,59,268,160)(14,60,269,161)(15,61,270,162)(16,62,226,163)(17,63,227,164)(18,64,228,165)(19,65,229,166)(20,66,230,167)(21,67,231,168)(22,68,232,169)(23,69,233,170)(24,70,234,171)(25,71,235,172)(26,72,236,173)(27,73,237,174)(28,74,238,175)(29,75,239,176)(30,76,240,177)(31,77,241,178)(32,78,242,179)(33,79,243,180)(34,80,244,136)(35,81,245,137)(36,82,246,138)(37,83,247,139)(38,84,248,140)(39,85,249,141)(40,86,250,142)(41,87,251,143)(42,88,252,144)(43,89,253,145)(44,90,254,146)(45,46,255,147)(91,191,345,271)(92,192,346,272)(93,193,347,273)(94,194,348,274)(95,195,349,275)(96,196,350,276)(97,197,351,277)(98,198,352,278)(99,199,353,279)(100,200,354,280)(101,201,355,281)(102,202,356,282)(103,203,357,283)(104,204,358,284)(105,205,359,285)(106,206,360,286)(107,207,316,287)(108,208,317,288)(109,209,318,289)(110,210,319,290)(111,211,320,291)(112,212,321,292)(113,213,322,293)(114,214,323,294)(115,215,324,295)(116,216,325,296)(117,217,326,297)(118,218,327,298)(119,219,328,299)(120,220,329,300)(121,221,330,301)(122,222,331,302)(123,223,332,303)(124,224,333,304)(125,225,334,305)(126,181,335,306)(127,182,336,307)(128,183,337,308)(129,184,338,309)(130,185,339,310)(131,186,340,311)(132,187,341,312)(133,188,342,313)(134,189,343,314)(135,190,344,315) );
G=PermutationGroup([[(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),(316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360)], [(1,284,256,204),(2,285,257,205),(3,286,258,206),(4,287,259,207),(5,288,260,208),(6,289,261,209),(7,290,262,210),(8,291,263,211),(9,292,264,212),(10,293,265,213),(11,294,266,214),(12,295,267,215),(13,296,268,216),(14,297,269,217),(15,298,270,218),(16,299,226,219),(17,300,227,220),(18,301,228,221),(19,302,229,222),(20,303,230,223),(21,304,231,224),(22,305,232,225),(23,306,233,181),(24,307,234,182),(25,308,235,183),(26,309,236,184),(27,310,237,185),(28,311,238,186),(29,312,239,187),(30,313,240,188),(31,314,241,189),(32,315,242,190),(33,271,243,191),(34,272,244,192),(35,273,245,193),(36,274,246,194),(37,275,247,195),(38,276,248,196),(39,277,249,197),(40,278,250,198),(41,279,251,199),(42,280,252,200),(43,281,253,201),(44,282,254,202),(45,283,255,203),(46,357,147,103),(47,358,148,104),(48,359,149,105),(49,360,150,106),(50,316,151,107),(51,317,152,108),(52,318,153,109),(53,319,154,110),(54,320,155,111),(55,321,156,112),(56,322,157,113),(57,323,158,114),(58,324,159,115),(59,325,160,116),(60,326,161,117),(61,327,162,118),(62,328,163,119),(63,329,164,120),(64,330,165,121),(65,331,166,122),(66,332,167,123),(67,333,168,124),(68,334,169,125),(69,335,170,126),(70,336,171,127),(71,337,172,128),(72,338,173,129),(73,339,174,130),(74,340,175,131),(75,341,176,132),(76,342,177,133),(77,343,178,134),(78,344,179,135),(79,345,180,91),(80,346,136,92),(81,347,137,93),(82,348,138,94),(83,349,139,95),(84,350,140,96),(85,351,141,97),(86,352,142,98),(87,353,143,99),(88,354,144,100),(89,355,145,101),(90,356,146,102)], [(1,47,256,148),(2,48,257,149),(3,49,258,150),(4,50,259,151),(5,51,260,152),(6,52,261,153),(7,53,262,154),(8,54,263,155),(9,55,264,156),(10,56,265,157),(11,57,266,158),(12,58,267,159),(13,59,268,160),(14,60,269,161),(15,61,270,162),(16,62,226,163),(17,63,227,164),(18,64,228,165),(19,65,229,166),(20,66,230,167),(21,67,231,168),(22,68,232,169),(23,69,233,170),(24,70,234,171),(25,71,235,172),(26,72,236,173),(27,73,237,174),(28,74,238,175),(29,75,239,176),(30,76,240,177),(31,77,241,178),(32,78,242,179),(33,79,243,180),(34,80,244,136),(35,81,245,137),(36,82,246,138),(37,83,247,139),(38,84,248,140),(39,85,249,141),(40,86,250,142),(41,87,251,143),(42,88,252,144),(43,89,253,145),(44,90,254,146),(45,46,255,147),(91,191,345,271),(92,192,346,272),(93,193,347,273),(94,194,348,274),(95,195,349,275),(96,196,350,276),(97,197,351,277),(98,198,352,278),(99,199,353,279),(100,200,354,280),(101,201,355,281),(102,202,356,282),(103,203,357,283),(104,204,358,284),(105,205,359,285),(106,206,360,286),(107,207,316,287),(108,208,317,288),(109,209,318,289),(110,210,319,290),(111,211,320,291),(112,212,321,292),(113,213,322,293),(114,214,323,294),(115,215,324,295),(116,216,325,296),(117,217,326,297),(118,218,327,298),(119,219,328,299),(120,220,329,300),(121,221,330,301),(122,222,331,302),(123,223,332,303),(124,224,333,304),(125,225,334,305),(126,181,335,306),(127,182,336,307),(128,183,337,308),(129,184,338,309),(130,185,339,310),(131,186,340,311),(132,187,341,312),(133,188,342,313),(134,189,343,314),(135,190,344,315)]])
225 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6A | 6B | 9A | ··· | 9F | 10A | 10B | 10C | 10D | 12A | ··· | 12F | 15A | ··· | 15H | 18A | ··· | 18F | 20A | ··· | 20L | 30A | ··· | 30H | 36A | ··· | 36R | 45A | ··· | 45X | 60A | ··· | 60X | 90A | ··· | 90X | 180A | ··· | 180BT |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 9 | ··· | 9 | 10 | 10 | 10 | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 18 | ··· | 18 | 20 | ··· | 20 | 30 | ··· | 30 | 36 | ··· | 36 | 45 | ··· | 45 | 60 | ··· | 60 | 90 | ··· | 90 | 180 | ··· | 180 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
225 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | - | |||||||||||||||
| image | C1 | C2 | C3 | C5 | C6 | C9 | C10 | C15 | C18 | C30 | C45 | C90 | Q8 | C3×Q8 | C5×Q8 | Q8×C9 | Q8×C15 | Q8×C45 |
| kernel | Q8×C45 | C180 | Q8×C15 | Q8×C9 | C60 | C5×Q8 | C36 | C3×Q8 | C20 | C12 | Q8 | C4 | C45 | C15 | C9 | C5 | C3 | C1 |
| # reps | 1 | 3 | 2 | 4 | 6 | 6 | 12 | 8 | 18 | 24 | 24 | 72 | 1 | 2 | 4 | 6 | 8 | 24 |
Matrix representation of Q8×C45 ►in GL3(𝔽181) generated by
| 73 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 5 |
| 1 | 0 | 0 |
| 0 | 1 | 179 |
| 0 | 1 | 180 |
| 180 | 0 | 0 |
| 0 | 110 | 66 |
| 0 | 143 | 71 |
G:=sub<GL(3,GF(181))| [73,0,0,0,5,0,0,0,5],[1,0,0,0,1,1,0,179,180],[180,0,0,0,110,143,0,66,71] >;
Q8×C45 in GAP, Magma, Sage, TeX
Q_8\times C_{45} % in TeX
G:=Group("Q8xC45"); // GroupNames label
G:=SmallGroup(360,32);
// by ID
G=gap.SmallGroup(360,32);
# by ID
G:=PCGroup([6,-2,-2,-3,-5,-2,-3,360,745,367,554]);
// Polycyclic
G:=Group<a,b,c|a^45=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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