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## G = C5×Dic18order 360 = 23·32·5

### Direct product of C5 and Dic18

Aliases: C5×Dic18, C453Q8, C60.8S3, C20.3D9, C180.3C2, C36.1C10, C30.56D6, Dic9.C10, C10.13D18, C15.3Dic6, C90.13C22, C9⋊(C5×Q8), C4.(C5×D9), C3.(C5×Dic6), C6.6(S3×C10), C12.1(C5×S3), C2.3(C10×D9), C18.1(C2×C10), (C5×Dic9).2C2, SmallGroup(360,20)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C5×Dic18
 Chief series C1 — C3 — C9 — C18 — C90 — C5×Dic9 — C5×Dic18
 Lower central C9 — C18 — C5×Dic18
 Upper central C1 — C10 — C20

Generators and relations for C5×Dic18
G = < a,b,c | a5=b36=1, c2=b18, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C5×Dic18
Regular action on 360 points
Generators in S360
(1 186 326 279 37)(2 187 327 280 38)(3 188 328 281 39)(4 189 329 282 40)(5 190 330 283 41)(6 191 331 284 42)(7 192 332 285 43)(8 193 333 286 44)(9 194 334 287 45)(10 195 335 288 46)(11 196 336 253 47)(12 197 337 254 48)(13 198 338 255 49)(14 199 339 256 50)(15 200 340 257 51)(16 201 341 258 52)(17 202 342 259 53)(18 203 343 260 54)(19 204 344 261 55)(20 205 345 262 56)(21 206 346 263 57)(22 207 347 264 58)(23 208 348 265 59)(24 209 349 266 60)(25 210 350 267 61)(26 211 351 268 62)(27 212 352 269 63)(28 213 353 270 64)(29 214 354 271 65)(30 215 355 272 66)(31 216 356 273 67)(32 181 357 274 68)(33 182 358 275 69)(34 183 359 276 70)(35 184 360 277 71)(36 185 325 278 72)(73 132 158 302 229)(74 133 159 303 230)(75 134 160 304 231)(76 135 161 305 232)(77 136 162 306 233)(78 137 163 307 234)(79 138 164 308 235)(80 139 165 309 236)(81 140 166 310 237)(82 141 167 311 238)(83 142 168 312 239)(84 143 169 313 240)(85 144 170 314 241)(86 109 171 315 242)(87 110 172 316 243)(88 111 173 317 244)(89 112 174 318 245)(90 113 175 319 246)(91 114 176 320 247)(92 115 177 321 248)(93 116 178 322 249)(94 117 179 323 250)(95 118 180 324 251)(96 119 145 289 252)(97 120 146 290 217)(98 121 147 291 218)(99 122 148 292 219)(100 123 149 293 220)(101 124 150 294 221)(102 125 151 295 222)(103 126 152 296 223)(104 127 153 297 224)(105 128 154 298 225)(106 129 155 299 226)(107 130 156 300 227)(108 131 157 301 228)
(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)
(1 313 19 295)(2 312 20 294)(3 311 21 293)(4 310 22 292)(5 309 23 291)(6 308 24 290)(7 307 25 289)(8 306 26 324)(9 305 27 323)(10 304 28 322)(11 303 29 321)(12 302 30 320)(13 301 31 319)(14 300 32 318)(15 299 33 317)(16 298 34 316)(17 297 35 315)(18 296 36 314)(37 169 55 151)(38 168 56 150)(39 167 57 149)(40 166 58 148)(41 165 59 147)(42 164 60 146)(43 163 61 145)(44 162 62 180)(45 161 63 179)(46 160 64 178)(47 159 65 177)(48 158 66 176)(49 157 67 175)(50 156 68 174)(51 155 69 173)(52 154 70 172)(53 153 71 171)(54 152 72 170)(73 355 91 337)(74 354 92 336)(75 353 93 335)(76 352 94 334)(77 351 95 333)(78 350 96 332)(79 349 97 331)(80 348 98 330)(81 347 99 329)(82 346 100 328)(83 345 101 327)(84 344 102 326)(85 343 103 325)(86 342 104 360)(87 341 105 359)(88 340 106 358)(89 339 107 357)(90 338 108 356)(109 259 127 277)(110 258 128 276)(111 257 129 275)(112 256 130 274)(113 255 131 273)(114 254 132 272)(115 253 133 271)(116 288 134 270)(117 287 135 269)(118 286 136 268)(119 285 137 267)(120 284 138 266)(121 283 139 265)(122 282 140 264)(123 281 141 263)(124 280 142 262)(125 279 143 261)(126 278 144 260)(181 245 199 227)(182 244 200 226)(183 243 201 225)(184 242 202 224)(185 241 203 223)(186 240 204 222)(187 239 205 221)(188 238 206 220)(189 237 207 219)(190 236 208 218)(191 235 209 217)(192 234 210 252)(193 233 211 251)(194 232 212 250)(195 231 213 249)(196 230 214 248)(197 229 215 247)(198 228 216 246)

G:=sub<Sym(360)| (1,186,326,279,37)(2,187,327,280,38)(3,188,328,281,39)(4,189,329,282,40)(5,190,330,283,41)(6,191,331,284,42)(7,192,332,285,43)(8,193,333,286,44)(9,194,334,287,45)(10,195,335,288,46)(11,196,336,253,47)(12,197,337,254,48)(13,198,338,255,49)(14,199,339,256,50)(15,200,340,257,51)(16,201,341,258,52)(17,202,342,259,53)(18,203,343,260,54)(19,204,344,261,55)(20,205,345,262,56)(21,206,346,263,57)(22,207,347,264,58)(23,208,348,265,59)(24,209,349,266,60)(25,210,350,267,61)(26,211,351,268,62)(27,212,352,269,63)(28,213,353,270,64)(29,214,354,271,65)(30,215,355,272,66)(31,216,356,273,67)(32,181,357,274,68)(33,182,358,275,69)(34,183,359,276,70)(35,184,360,277,71)(36,185,325,278,72)(73,132,158,302,229)(74,133,159,303,230)(75,134,160,304,231)(76,135,161,305,232)(77,136,162,306,233)(78,137,163,307,234)(79,138,164,308,235)(80,139,165,309,236)(81,140,166,310,237)(82,141,167,311,238)(83,142,168,312,239)(84,143,169,313,240)(85,144,170,314,241)(86,109,171,315,242)(87,110,172,316,243)(88,111,173,317,244)(89,112,174,318,245)(90,113,175,319,246)(91,114,176,320,247)(92,115,177,321,248)(93,116,178,322,249)(94,117,179,323,250)(95,118,180,324,251)(96,119,145,289,252)(97,120,146,290,217)(98,121,147,291,218)(99,122,148,292,219)(100,123,149,293,220)(101,124,150,294,221)(102,125,151,295,222)(103,126,152,296,223)(104,127,153,297,224)(105,128,154,298,225)(106,129,155,299,226)(107,130,156,300,227)(108,131,157,301,228), (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), (1,313,19,295)(2,312,20,294)(3,311,21,293)(4,310,22,292)(5,309,23,291)(6,308,24,290)(7,307,25,289)(8,306,26,324)(9,305,27,323)(10,304,28,322)(11,303,29,321)(12,302,30,320)(13,301,31,319)(14,300,32,318)(15,299,33,317)(16,298,34,316)(17,297,35,315)(18,296,36,314)(37,169,55,151)(38,168,56,150)(39,167,57,149)(40,166,58,148)(41,165,59,147)(42,164,60,146)(43,163,61,145)(44,162,62,180)(45,161,63,179)(46,160,64,178)(47,159,65,177)(48,158,66,176)(49,157,67,175)(50,156,68,174)(51,155,69,173)(52,154,70,172)(53,153,71,171)(54,152,72,170)(73,355,91,337)(74,354,92,336)(75,353,93,335)(76,352,94,334)(77,351,95,333)(78,350,96,332)(79,349,97,331)(80,348,98,330)(81,347,99,329)(82,346,100,328)(83,345,101,327)(84,344,102,326)(85,343,103,325)(86,342,104,360)(87,341,105,359)(88,340,106,358)(89,339,107,357)(90,338,108,356)(109,259,127,277)(110,258,128,276)(111,257,129,275)(112,256,130,274)(113,255,131,273)(114,254,132,272)(115,253,133,271)(116,288,134,270)(117,287,135,269)(118,286,136,268)(119,285,137,267)(120,284,138,266)(121,283,139,265)(122,282,140,264)(123,281,141,263)(124,280,142,262)(125,279,143,261)(126,278,144,260)(181,245,199,227)(182,244,200,226)(183,243,201,225)(184,242,202,224)(185,241,203,223)(186,240,204,222)(187,239,205,221)(188,238,206,220)(189,237,207,219)(190,236,208,218)(191,235,209,217)(192,234,210,252)(193,233,211,251)(194,232,212,250)(195,231,213,249)(196,230,214,248)(197,229,215,247)(198,228,216,246)>;

G:=Group( (1,186,326,279,37)(2,187,327,280,38)(3,188,328,281,39)(4,189,329,282,40)(5,190,330,283,41)(6,191,331,284,42)(7,192,332,285,43)(8,193,333,286,44)(9,194,334,287,45)(10,195,335,288,46)(11,196,336,253,47)(12,197,337,254,48)(13,198,338,255,49)(14,199,339,256,50)(15,200,340,257,51)(16,201,341,258,52)(17,202,342,259,53)(18,203,343,260,54)(19,204,344,261,55)(20,205,345,262,56)(21,206,346,263,57)(22,207,347,264,58)(23,208,348,265,59)(24,209,349,266,60)(25,210,350,267,61)(26,211,351,268,62)(27,212,352,269,63)(28,213,353,270,64)(29,214,354,271,65)(30,215,355,272,66)(31,216,356,273,67)(32,181,357,274,68)(33,182,358,275,69)(34,183,359,276,70)(35,184,360,277,71)(36,185,325,278,72)(73,132,158,302,229)(74,133,159,303,230)(75,134,160,304,231)(76,135,161,305,232)(77,136,162,306,233)(78,137,163,307,234)(79,138,164,308,235)(80,139,165,309,236)(81,140,166,310,237)(82,141,167,311,238)(83,142,168,312,239)(84,143,169,313,240)(85,144,170,314,241)(86,109,171,315,242)(87,110,172,316,243)(88,111,173,317,244)(89,112,174,318,245)(90,113,175,319,246)(91,114,176,320,247)(92,115,177,321,248)(93,116,178,322,249)(94,117,179,323,250)(95,118,180,324,251)(96,119,145,289,252)(97,120,146,290,217)(98,121,147,291,218)(99,122,148,292,219)(100,123,149,293,220)(101,124,150,294,221)(102,125,151,295,222)(103,126,152,296,223)(104,127,153,297,224)(105,128,154,298,225)(106,129,155,299,226)(107,130,156,300,227)(108,131,157,301,228), (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), (1,313,19,295)(2,312,20,294)(3,311,21,293)(4,310,22,292)(5,309,23,291)(6,308,24,290)(7,307,25,289)(8,306,26,324)(9,305,27,323)(10,304,28,322)(11,303,29,321)(12,302,30,320)(13,301,31,319)(14,300,32,318)(15,299,33,317)(16,298,34,316)(17,297,35,315)(18,296,36,314)(37,169,55,151)(38,168,56,150)(39,167,57,149)(40,166,58,148)(41,165,59,147)(42,164,60,146)(43,163,61,145)(44,162,62,180)(45,161,63,179)(46,160,64,178)(47,159,65,177)(48,158,66,176)(49,157,67,175)(50,156,68,174)(51,155,69,173)(52,154,70,172)(53,153,71,171)(54,152,72,170)(73,355,91,337)(74,354,92,336)(75,353,93,335)(76,352,94,334)(77,351,95,333)(78,350,96,332)(79,349,97,331)(80,348,98,330)(81,347,99,329)(82,346,100,328)(83,345,101,327)(84,344,102,326)(85,343,103,325)(86,342,104,360)(87,341,105,359)(88,340,106,358)(89,339,107,357)(90,338,108,356)(109,259,127,277)(110,258,128,276)(111,257,129,275)(112,256,130,274)(113,255,131,273)(114,254,132,272)(115,253,133,271)(116,288,134,270)(117,287,135,269)(118,286,136,268)(119,285,137,267)(120,284,138,266)(121,283,139,265)(122,282,140,264)(123,281,141,263)(124,280,142,262)(125,279,143,261)(126,278,144,260)(181,245,199,227)(182,244,200,226)(183,243,201,225)(184,242,202,224)(185,241,203,223)(186,240,204,222)(187,239,205,221)(188,238,206,220)(189,237,207,219)(190,236,208,218)(191,235,209,217)(192,234,210,252)(193,233,211,251)(194,232,212,250)(195,231,213,249)(196,230,214,248)(197,229,215,247)(198,228,216,246) );

G=PermutationGroup([[(1,186,326,279,37),(2,187,327,280,38),(3,188,328,281,39),(4,189,329,282,40),(5,190,330,283,41),(6,191,331,284,42),(7,192,332,285,43),(8,193,333,286,44),(9,194,334,287,45),(10,195,335,288,46),(11,196,336,253,47),(12,197,337,254,48),(13,198,338,255,49),(14,199,339,256,50),(15,200,340,257,51),(16,201,341,258,52),(17,202,342,259,53),(18,203,343,260,54),(19,204,344,261,55),(20,205,345,262,56),(21,206,346,263,57),(22,207,347,264,58),(23,208,348,265,59),(24,209,349,266,60),(25,210,350,267,61),(26,211,351,268,62),(27,212,352,269,63),(28,213,353,270,64),(29,214,354,271,65),(30,215,355,272,66),(31,216,356,273,67),(32,181,357,274,68),(33,182,358,275,69),(34,183,359,276,70),(35,184,360,277,71),(36,185,325,278,72),(73,132,158,302,229),(74,133,159,303,230),(75,134,160,304,231),(76,135,161,305,232),(77,136,162,306,233),(78,137,163,307,234),(79,138,164,308,235),(80,139,165,309,236),(81,140,166,310,237),(82,141,167,311,238),(83,142,168,312,239),(84,143,169,313,240),(85,144,170,314,241),(86,109,171,315,242),(87,110,172,316,243),(88,111,173,317,244),(89,112,174,318,245),(90,113,175,319,246),(91,114,176,320,247),(92,115,177,321,248),(93,116,178,322,249),(94,117,179,323,250),(95,118,180,324,251),(96,119,145,289,252),(97,120,146,290,217),(98,121,147,291,218),(99,122,148,292,219),(100,123,149,293,220),(101,124,150,294,221),(102,125,151,295,222),(103,126,152,296,223),(104,127,153,297,224),(105,128,154,298,225),(106,129,155,299,226),(107,130,156,300,227),(108,131,157,301,228)], [(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)], [(1,313,19,295),(2,312,20,294),(3,311,21,293),(4,310,22,292),(5,309,23,291),(6,308,24,290),(7,307,25,289),(8,306,26,324),(9,305,27,323),(10,304,28,322),(11,303,29,321),(12,302,30,320),(13,301,31,319),(14,300,32,318),(15,299,33,317),(16,298,34,316),(17,297,35,315),(18,296,36,314),(37,169,55,151),(38,168,56,150),(39,167,57,149),(40,166,58,148),(41,165,59,147),(42,164,60,146),(43,163,61,145),(44,162,62,180),(45,161,63,179),(46,160,64,178),(47,159,65,177),(48,158,66,176),(49,157,67,175),(50,156,68,174),(51,155,69,173),(52,154,70,172),(53,153,71,171),(54,152,72,170),(73,355,91,337),(74,354,92,336),(75,353,93,335),(76,352,94,334),(77,351,95,333),(78,350,96,332),(79,349,97,331),(80,348,98,330),(81,347,99,329),(82,346,100,328),(83,345,101,327),(84,344,102,326),(85,343,103,325),(86,342,104,360),(87,341,105,359),(88,340,106,358),(89,339,107,357),(90,338,108,356),(109,259,127,277),(110,258,128,276),(111,257,129,275),(112,256,130,274),(113,255,131,273),(114,254,132,272),(115,253,133,271),(116,288,134,270),(117,287,135,269),(118,286,136,268),(119,285,137,267),(120,284,138,266),(121,283,139,265),(122,282,140,264),(123,281,141,263),(124,280,142,262),(125,279,143,261),(126,278,144,260),(181,245,199,227),(182,244,200,226),(183,243,201,225),(184,242,202,224),(185,241,203,223),(186,240,204,222),(187,239,205,221),(188,238,206,220),(189,237,207,219),(190,236,208,218),(191,235,209,217),(192,234,210,252),(193,233,211,251),(194,232,212,250),(195,231,213,249),(196,230,214,248),(197,229,215,247),(198,228,216,246)]])

105 conjugacy classes

 class 1 2 3 4A 4B 4C 5A 5B 5C 5D 6 9A 9B 9C 10A 10B 10C 10D 12A 12B 15A 15B 15C 15D 18A 18B 18C 20A 20B 20C 20D 20E ··· 20L 30A 30B 30C 30D 36A ··· 36F 45A ··· 45L 60A ··· 60H 90A ··· 90L 180A ··· 180X order 1 2 3 4 4 4 5 5 5 5 6 9 9 9 10 10 10 10 12 12 15 15 15 15 18 18 18 20 20 20 20 20 ··· 20 30 30 30 30 36 ··· 36 45 ··· 45 60 ··· 60 90 ··· 90 180 ··· 180 size 1 1 2 2 18 18 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 18 ··· 18 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

105 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + + - + - image C1 C2 C2 C5 C10 C10 S3 Q8 D6 D9 Dic6 C5×S3 D18 C5×Q8 S3×C10 Dic18 C5×D9 C5×Dic6 C10×D9 C5×Dic18 kernel C5×Dic18 C5×Dic9 C180 Dic18 Dic9 C36 C60 C45 C30 C20 C15 C12 C10 C9 C6 C5 C4 C3 C2 C1 # reps 1 2 1 4 8 4 1 1 1 3 2 4 3 4 4 6 12 8 12 24

Matrix representation of C5×Dic18 in GL2(𝔽181) generated by

 59 0 0 59
,
 58 144 37 95
,
 69 158 89 112
G:=sub<GL(2,GF(181))| [59,0,0,59],[58,37,144,95],[69,89,158,112] >;

C5×Dic18 in GAP, Magma, Sage, TeX

C_5\times {\rm Dic}_{18}
% in TeX

G:=Group("C5xDic18");
// GroupNames label

G:=SmallGroup(360,20);
// by ID

G=gap.SmallGroup(360,20);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-3,-3,120,265,127,6004,208,8645]);
// Polycyclic

G:=Group<a,b,c|a^5=b^36=1,c^2=b^18,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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