metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C45⋊Q8, C30.1D6, C5⋊1Dic18, C9⋊1Dic10, Dic9.D5, C15.Dic6, C18.1D10, C10.1D18, C90.1C22, Dic5.1D9, Dic45.2C2, C3.(C15⋊Q8), C6.8(S3×D5), C2.4(D5×D9), (C5×Dic9).1C2, (C3×Dic5).1S3, (C9×Dic5).1C2, SmallGroup(360,7)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C45⋊Q8
G = < a,b,c | a45=b4=1, c2=b2, bab-1=a26, cac-1=a19, cbc-1=b-1 >
Smallest permutation representation of C45⋊Q8
►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)
(1 165 46 91)(2 146 47 117)(3 172 48 98)(4 153 49 124)(5 179 50 105)(6 160 51 131)(7 141 52 112)(8 167 53 93)(9 148 54 119)(10 174 55 100)(11 155 56 126)(12 136 57 107)(13 162 58 133)(14 143 59 114)(15 169 60 95)(16 150 61 121)(17 176 62 102)(18 157 63 128)(19 138 64 109)(20 164 65 135)(21 145 66 116)(22 171 67 97)(23 152 68 123)(24 178 69 104)(25 159 70 130)(26 140 71 111)(27 166 72 92)(28 147 73 118)(29 173 74 99)(30 154 75 125)(31 180 76 106)(32 161 77 132)(33 142 78 113)(34 168 79 94)(35 149 80 120)(36 175 81 101)(37 156 82 127)(38 137 83 108)(39 163 84 134)(40 144 85 115)(41 170 86 96)(42 151 87 122)(43 177 88 103)(44 158 89 129)(45 139 90 110)(181 271 254 359)(182 297 255 340)(183 278 256 321)(184 304 257 347)(185 285 258 328)(186 311 259 354)(187 292 260 335)(188 273 261 316)(189 299 262 342)(190 280 263 323)(191 306 264 349)(192 287 265 330)(193 313 266 356)(194 294 267 337)(195 275 268 318)(196 301 269 344)(197 282 270 325)(198 308 226 351)(199 289 227 332)(200 315 228 358)(201 296 229 339)(202 277 230 320)(203 303 231 346)(204 284 232 327)(205 310 233 353)(206 291 234 334)(207 272 235 360)(208 298 236 341)(209 279 237 322)(210 305 238 348)(211 286 239 329)(212 312 240 355)(213 293 241 336)(214 274 242 317)(215 300 243 343)(216 281 244 324)(217 307 245 350)(218 288 246 331)(219 314 247 357)(220 295 248 338)(221 276 249 319)(222 302 250 345)(223 283 251 326)(224 309 252 352)(225 290 253 333)
(1 262 46 189)(2 236 47 208)(3 255 48 182)(4 229 49 201)(5 248 50 220)(6 267 51 194)(7 241 52 213)(8 260 53 187)(9 234 54 206)(10 253 55 225)(11 227 56 199)(12 246 57 218)(13 265 58 192)(14 239 59 211)(15 258 60 185)(16 232 61 204)(17 251 62 223)(18 270 63 197)(19 244 64 216)(20 263 65 190)(21 237 66 209)(22 256 67 183)(23 230 68 202)(24 249 69 221)(25 268 70 195)(26 242 71 214)(27 261 72 188)(28 235 73 207)(29 254 74 181)(30 228 75 200)(31 247 76 219)(32 266 77 193)(33 240 78 212)(34 259 79 186)(35 233 80 205)(36 252 81 224)(37 226 82 198)(38 245 83 217)(39 264 84 191)(40 238 85 210)(41 257 86 184)(42 231 87 203)(43 250 88 222)(44 269 89 196)(45 243 90 215)(91 342 165 299)(92 316 166 273)(93 335 167 292)(94 354 168 311)(95 328 169 285)(96 347 170 304)(97 321 171 278)(98 340 172 297)(99 359 173 271)(100 333 174 290)(101 352 175 309)(102 326 176 283)(103 345 177 302)(104 319 178 276)(105 338 179 295)(106 357 180 314)(107 331 136 288)(108 350 137 307)(109 324 138 281)(110 343 139 300)(111 317 140 274)(112 336 141 293)(113 355 142 312)(114 329 143 286)(115 348 144 305)(116 322 145 279)(117 341 146 298)(118 360 147 272)(119 334 148 291)(120 353 149 310)(121 327 150 284)(122 346 151 303)(123 320 152 277)(124 339 153 296)(125 358 154 315)(126 332 155 289)(127 351 156 308)(128 325 157 282)(129 344 158 301)(130 318 159 275)(131 337 160 294)(132 356 161 313)(133 330 162 287)(134 349 163 306)(135 323 164 280)
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), (1,165,46,91)(2,146,47,117)(3,172,48,98)(4,153,49,124)(5,179,50,105)(6,160,51,131)(7,141,52,112)(8,167,53,93)(9,148,54,119)(10,174,55,100)(11,155,56,126)(12,136,57,107)(13,162,58,133)(14,143,59,114)(15,169,60,95)(16,150,61,121)(17,176,62,102)(18,157,63,128)(19,138,64,109)(20,164,65,135)(21,145,66,116)(22,171,67,97)(23,152,68,123)(24,178,69,104)(25,159,70,130)(26,140,71,111)(27,166,72,92)(28,147,73,118)(29,173,74,99)(30,154,75,125)(31,180,76,106)(32,161,77,132)(33,142,78,113)(34,168,79,94)(35,149,80,120)(36,175,81,101)(37,156,82,127)(38,137,83,108)(39,163,84,134)(40,144,85,115)(41,170,86,96)(42,151,87,122)(43,177,88,103)(44,158,89,129)(45,139,90,110)(181,271,254,359)(182,297,255,340)(183,278,256,321)(184,304,257,347)(185,285,258,328)(186,311,259,354)(187,292,260,335)(188,273,261,316)(189,299,262,342)(190,280,263,323)(191,306,264,349)(192,287,265,330)(193,313,266,356)(194,294,267,337)(195,275,268,318)(196,301,269,344)(197,282,270,325)(198,308,226,351)(199,289,227,332)(200,315,228,358)(201,296,229,339)(202,277,230,320)(203,303,231,346)(204,284,232,327)(205,310,233,353)(206,291,234,334)(207,272,235,360)(208,298,236,341)(209,279,237,322)(210,305,238,348)(211,286,239,329)(212,312,240,355)(213,293,241,336)(214,274,242,317)(215,300,243,343)(216,281,244,324)(217,307,245,350)(218,288,246,331)(219,314,247,357)(220,295,248,338)(221,276,249,319)(222,302,250,345)(223,283,251,326)(224,309,252,352)(225,290,253,333), (1,262,46,189)(2,236,47,208)(3,255,48,182)(4,229,49,201)(5,248,50,220)(6,267,51,194)(7,241,52,213)(8,260,53,187)(9,234,54,206)(10,253,55,225)(11,227,56,199)(12,246,57,218)(13,265,58,192)(14,239,59,211)(15,258,60,185)(16,232,61,204)(17,251,62,223)(18,270,63,197)(19,244,64,216)(20,263,65,190)(21,237,66,209)(22,256,67,183)(23,230,68,202)(24,249,69,221)(25,268,70,195)(26,242,71,214)(27,261,72,188)(28,235,73,207)(29,254,74,181)(30,228,75,200)(31,247,76,219)(32,266,77,193)(33,240,78,212)(34,259,79,186)(35,233,80,205)(36,252,81,224)(37,226,82,198)(38,245,83,217)(39,264,84,191)(40,238,85,210)(41,257,86,184)(42,231,87,203)(43,250,88,222)(44,269,89,196)(45,243,90,215)(91,342,165,299)(92,316,166,273)(93,335,167,292)(94,354,168,311)(95,328,169,285)(96,347,170,304)(97,321,171,278)(98,340,172,297)(99,359,173,271)(100,333,174,290)(101,352,175,309)(102,326,176,283)(103,345,177,302)(104,319,178,276)(105,338,179,295)(106,357,180,314)(107,331,136,288)(108,350,137,307)(109,324,138,281)(110,343,139,300)(111,317,140,274)(112,336,141,293)(113,355,142,312)(114,329,143,286)(115,348,144,305)(116,322,145,279)(117,341,146,298)(118,360,147,272)(119,334,148,291)(120,353,149,310)(121,327,150,284)(122,346,151,303)(123,320,152,277)(124,339,153,296)(125,358,154,315)(126,332,155,289)(127,351,156,308)(128,325,157,282)(129,344,158,301)(130,318,159,275)(131,337,160,294)(132,356,161,313)(133,330,162,287)(134,349,163,306)(135,323,164,280)>;
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), (1,165,46,91)(2,146,47,117)(3,172,48,98)(4,153,49,124)(5,179,50,105)(6,160,51,131)(7,141,52,112)(8,167,53,93)(9,148,54,119)(10,174,55,100)(11,155,56,126)(12,136,57,107)(13,162,58,133)(14,143,59,114)(15,169,60,95)(16,150,61,121)(17,176,62,102)(18,157,63,128)(19,138,64,109)(20,164,65,135)(21,145,66,116)(22,171,67,97)(23,152,68,123)(24,178,69,104)(25,159,70,130)(26,140,71,111)(27,166,72,92)(28,147,73,118)(29,173,74,99)(30,154,75,125)(31,180,76,106)(32,161,77,132)(33,142,78,113)(34,168,79,94)(35,149,80,120)(36,175,81,101)(37,156,82,127)(38,137,83,108)(39,163,84,134)(40,144,85,115)(41,170,86,96)(42,151,87,122)(43,177,88,103)(44,158,89,129)(45,139,90,110)(181,271,254,359)(182,297,255,340)(183,278,256,321)(184,304,257,347)(185,285,258,328)(186,311,259,354)(187,292,260,335)(188,273,261,316)(189,299,262,342)(190,280,263,323)(191,306,264,349)(192,287,265,330)(193,313,266,356)(194,294,267,337)(195,275,268,318)(196,301,269,344)(197,282,270,325)(198,308,226,351)(199,289,227,332)(200,315,228,358)(201,296,229,339)(202,277,230,320)(203,303,231,346)(204,284,232,327)(205,310,233,353)(206,291,234,334)(207,272,235,360)(208,298,236,341)(209,279,237,322)(210,305,238,348)(211,286,239,329)(212,312,240,355)(213,293,241,336)(214,274,242,317)(215,300,243,343)(216,281,244,324)(217,307,245,350)(218,288,246,331)(219,314,247,357)(220,295,248,338)(221,276,249,319)(222,302,250,345)(223,283,251,326)(224,309,252,352)(225,290,253,333), (1,262,46,189)(2,236,47,208)(3,255,48,182)(4,229,49,201)(5,248,50,220)(6,267,51,194)(7,241,52,213)(8,260,53,187)(9,234,54,206)(10,253,55,225)(11,227,56,199)(12,246,57,218)(13,265,58,192)(14,239,59,211)(15,258,60,185)(16,232,61,204)(17,251,62,223)(18,270,63,197)(19,244,64,216)(20,263,65,190)(21,237,66,209)(22,256,67,183)(23,230,68,202)(24,249,69,221)(25,268,70,195)(26,242,71,214)(27,261,72,188)(28,235,73,207)(29,254,74,181)(30,228,75,200)(31,247,76,219)(32,266,77,193)(33,240,78,212)(34,259,79,186)(35,233,80,205)(36,252,81,224)(37,226,82,198)(38,245,83,217)(39,264,84,191)(40,238,85,210)(41,257,86,184)(42,231,87,203)(43,250,88,222)(44,269,89,196)(45,243,90,215)(91,342,165,299)(92,316,166,273)(93,335,167,292)(94,354,168,311)(95,328,169,285)(96,347,170,304)(97,321,171,278)(98,340,172,297)(99,359,173,271)(100,333,174,290)(101,352,175,309)(102,326,176,283)(103,345,177,302)(104,319,178,276)(105,338,179,295)(106,357,180,314)(107,331,136,288)(108,350,137,307)(109,324,138,281)(110,343,139,300)(111,317,140,274)(112,336,141,293)(113,355,142,312)(114,329,143,286)(115,348,144,305)(116,322,145,279)(117,341,146,298)(118,360,147,272)(119,334,148,291)(120,353,149,310)(121,327,150,284)(122,346,151,303)(123,320,152,277)(124,339,153,296)(125,358,154,315)(126,332,155,289)(127,351,156,308)(128,325,157,282)(129,344,158,301)(130,318,159,275)(131,337,160,294)(132,356,161,313)(133,330,162,287)(134,349,163,306)(135,323,164,280) );
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)], [(1,165,46,91),(2,146,47,117),(3,172,48,98),(4,153,49,124),(5,179,50,105),(6,160,51,131),(7,141,52,112),(8,167,53,93),(9,148,54,119),(10,174,55,100),(11,155,56,126),(12,136,57,107),(13,162,58,133),(14,143,59,114),(15,169,60,95),(16,150,61,121),(17,176,62,102),(18,157,63,128),(19,138,64,109),(20,164,65,135),(21,145,66,116),(22,171,67,97),(23,152,68,123),(24,178,69,104),(25,159,70,130),(26,140,71,111),(27,166,72,92),(28,147,73,118),(29,173,74,99),(30,154,75,125),(31,180,76,106),(32,161,77,132),(33,142,78,113),(34,168,79,94),(35,149,80,120),(36,175,81,101),(37,156,82,127),(38,137,83,108),(39,163,84,134),(40,144,85,115),(41,170,86,96),(42,151,87,122),(43,177,88,103),(44,158,89,129),(45,139,90,110),(181,271,254,359),(182,297,255,340),(183,278,256,321),(184,304,257,347),(185,285,258,328),(186,311,259,354),(187,292,260,335),(188,273,261,316),(189,299,262,342),(190,280,263,323),(191,306,264,349),(192,287,265,330),(193,313,266,356),(194,294,267,337),(195,275,268,318),(196,301,269,344),(197,282,270,325),(198,308,226,351),(199,289,227,332),(200,315,228,358),(201,296,229,339),(202,277,230,320),(203,303,231,346),(204,284,232,327),(205,310,233,353),(206,291,234,334),(207,272,235,360),(208,298,236,341),(209,279,237,322),(210,305,238,348),(211,286,239,329),(212,312,240,355),(213,293,241,336),(214,274,242,317),(215,300,243,343),(216,281,244,324),(217,307,245,350),(218,288,246,331),(219,314,247,357),(220,295,248,338),(221,276,249,319),(222,302,250,345),(223,283,251,326),(224,309,252,352),(225,290,253,333)], [(1,262,46,189),(2,236,47,208),(3,255,48,182),(4,229,49,201),(5,248,50,220),(6,267,51,194),(7,241,52,213),(8,260,53,187),(9,234,54,206),(10,253,55,225),(11,227,56,199),(12,246,57,218),(13,265,58,192),(14,239,59,211),(15,258,60,185),(16,232,61,204),(17,251,62,223),(18,270,63,197),(19,244,64,216),(20,263,65,190),(21,237,66,209),(22,256,67,183),(23,230,68,202),(24,249,69,221),(25,268,70,195),(26,242,71,214),(27,261,72,188),(28,235,73,207),(29,254,74,181),(30,228,75,200),(31,247,76,219),(32,266,77,193),(33,240,78,212),(34,259,79,186),(35,233,80,205),(36,252,81,224),(37,226,82,198),(38,245,83,217),(39,264,84,191),(40,238,85,210),(41,257,86,184),(42,231,87,203),(43,250,88,222),(44,269,89,196),(45,243,90,215),(91,342,165,299),(92,316,166,273),(93,335,167,292),(94,354,168,311),(95,328,169,285),(96,347,170,304),(97,321,171,278),(98,340,172,297),(99,359,173,271),(100,333,174,290),(101,352,175,309),(102,326,176,283),(103,345,177,302),(104,319,178,276),(105,338,179,295),(106,357,180,314),(107,331,136,288),(108,350,137,307),(109,324,138,281),(110,343,139,300),(111,317,140,274),(112,336,141,293),(113,355,142,312),(114,329,143,286),(115,348,144,305),(116,322,145,279),(117,341,146,298),(118,360,147,272),(119,334,148,291),(120,353,149,310),(121,327,150,284),(122,346,151,303),(123,320,152,277),(124,339,153,296),(125,358,154,315),(126,332,155,289),(127,351,156,308),(128,325,157,282),(129,344,158,301),(130,318,159,275),(131,337,160,294),(132,356,161,313),(133,330,162,287),(134,349,163,306),(135,323,164,280)]])
45 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 5A | 5B | 6 | 9A | 9B | 9C | 10A | 10B | 12A | 12B | 15A | 15B | 18A | 18B | 18C | 20A | 20B | 20C | 20D | 30A | 30B | 36A | ··· | 36F | 45A | ··· | 45F | 90A | ··· | 90F |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 9 | 9 | 9 | 10 | 10 | 12 | 12 | 15 | 15 | 18 | 18 | 18 | 20 | 20 | 20 | 20 | 30 | 30 | 36 | ··· | 36 | 45 | ··· | 45 | 90 | ··· | 90 |
| size | 1 | 1 | 2 | 10 | 18 | 90 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 4 | 4 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 4 | 4 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | + | - | + | - | - | + | - | + | - |
| image | C1 | C2 | C2 | C2 | S3 | Q8 | D5 | D6 | D9 | D10 | Dic6 | D18 | Dic10 | Dic18 | S3×D5 | C15⋊Q8 | D5×D9 | C45⋊Q8 |
| kernel | C45⋊Q8 | C5×Dic9 | C9×Dic5 | Dic45 | C3×Dic5 | C45 | Dic9 | C30 | Dic5 | C18 | C15 | C10 | C9 | C5 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 3 | 2 | 2 | 3 | 4 | 6 | 2 | 2 | 6 | 6 |
Matrix representation of C45⋊Q8 ►in GL4(𝔽181) generated by
| 14 | 180 | 0 | 0 |
| 15 | 180 | 0 | 0 |
| 0 | 0 | 131 | 54 |
| 0 | 0 | 127 | 4 |
| 132 | 91 | 0 | 0 |
| 83 | 49 | 0 | 0 |
| 0 | 0 | 144 | 123 |
| 0 | 0 | 86 | 37 |
| 108 | 25 | 0 | 0 |
| 4 | 73 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(181))| [14,15,0,0,180,180,0,0,0,0,131,127,0,0,54,4],[132,83,0,0,91,49,0,0,0,0,144,86,0,0,123,37],[108,4,0,0,25,73,0,0,0,0,1,0,0,0,0,1] >;
C45⋊Q8 in GAP, Magma, Sage, TeX
C_{45}\rtimes Q_8 % in TeX
G:=Group("C45:Q8"); // GroupNames label
G:=SmallGroup(360,7);
// by ID
G=gap.SmallGroup(360,7);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-5,-3,24,73,31,1641,741,2884,4331]);
// Polycyclic
G:=Group<a,b,c|a^45=b^4=1,c^2=b^2,b*a*b^-1=a^26,c*a*c^-1=a^19,c*b*c^-1=b^-1>;
// generators/relations
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