metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.88D4, C40.17Q8, Dic5⋊1C16, C8.16Dic10, C10.7M5(2), C20.39M4(2), C5⋊4(C4⋊C16), (C2×C80).1C2, C2.4(D5×C16), (C2×C16).1D5, C10.17(C4⋊C8), C20.71(C4⋊C4), C22.9(C8×D5), C10.14(C2×C16), (C2×C8).333D10, C8.48(C5⋊D4), (C2×Dic5).3C8, (C8×Dic5).8C2, C2.1(C80⋊C2), C4.14(C8⋊D5), (C4×Dic5).13C4, (C2×C40).399C22, C2.1(C20.8Q8), C4.27(C10.D4), (C2×C5⋊2C16).9C2, (C2×C5⋊2C8).15C4, (C2×C10).38(C2×C8), (C2×C4).167(C4×D5), (C2×C20).414(C2×C4), SmallGroup(320,59)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.88D4
G = < a,b,c | a40=b4=1, c2=a25, bab-1=cac-1=a9, cbc-1=b-1 >
Smallest permutation representation of C40.88D4
►Regular action on 320 points
Generators in S320
(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)
(1 54 225 148)(2 63 226 157)(3 72 227 126)(4 41 228 135)(5 50 229 144)(6 59 230 153)(7 68 231 122)(8 77 232 131)(9 46 233 140)(10 55 234 149)(11 64 235 158)(12 73 236 127)(13 42 237 136)(14 51 238 145)(15 60 239 154)(16 69 240 123)(17 78 201 132)(18 47 202 141)(19 56 203 150)(20 65 204 159)(21 74 205 128)(22 43 206 137)(23 52 207 146)(24 61 208 155)(25 70 209 124)(26 79 210 133)(27 48 211 142)(28 57 212 151)(29 66 213 160)(30 75 214 129)(31 44 215 138)(32 53 216 147)(33 62 217 156)(34 71 218 125)(35 80 219 134)(36 49 220 143)(37 58 221 152)(38 67 222 121)(39 76 223 130)(40 45 224 139)(81 248 193 314)(82 257 194 283)(83 266 195 292)(84 275 196 301)(85 244 197 310)(86 253 198 319)(87 262 199 288)(88 271 200 297)(89 280 161 306)(90 249 162 315)(91 258 163 284)(92 267 164 293)(93 276 165 302)(94 245 166 311)(95 254 167 320)(96 263 168 289)(97 272 169 298)(98 241 170 307)(99 250 171 316)(100 259 172 285)(101 268 173 294)(102 277 174 303)(103 246 175 312)(104 255 176 281)(105 264 177 290)(106 273 178 299)(107 242 179 308)(108 251 180 317)(109 260 181 286)(110 269 182 295)(111 278 183 304)(112 247 184 313)(113 256 185 282)(114 265 186 291)(115 274 187 300)(116 243 188 309)(117 252 189 318)(118 261 190 287)(119 270 191 296)(120 279 192 305)
(1 101 26 86 11 111 36 96 21 81 6 106 31 91 16 116)(2 110 27 95 12 120 37 105 22 90 7 115 32 100 17 85)(3 119 28 104 13 89 38 114 23 99 8 84 33 109 18 94)(4 88 29 113 14 98 39 83 24 108 9 93 34 118 19 103)(5 97 30 82 15 107 40 92 25 117 10 102 35 87 20 112)(41 297 66 282 51 307 76 292 61 317 46 302 71 287 56 312)(42 306 67 291 52 316 77 301 62 286 47 311 72 296 57 281)(43 315 68 300 53 285 78 310 63 295 48 320 73 305 58 290)(44 284 69 309 54 294 79 319 64 304 49 289 74 314 59 299)(45 293 70 318 55 303 80 288 65 313 50 298 75 283 60 308)(121 265 146 250 131 275 156 260 141 245 126 270 151 255 136 280)(122 274 147 259 132 244 157 269 142 254 127 279 152 264 137 249)(123 243 148 268 133 253 158 278 143 263 128 248 153 273 138 258)(124 252 149 277 134 262 159 247 144 272 129 257 154 242 139 267)(125 261 150 246 135 271 160 256 145 241 130 266 155 251 140 276)(161 222 186 207 171 232 196 217 181 202 166 227 191 212 176 237)(162 231 187 216 172 201 197 226 182 211 167 236 192 221 177 206)(163 240 188 225 173 210 198 235 183 220 168 205 193 230 178 215)(164 209 189 234 174 219 199 204 184 229 169 214 194 239 179 224)(165 218 190 203 175 228 200 213 185 238 170 223 195 208 180 233)
G:=sub<Sym(320)| (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), (1,54,225,148)(2,63,226,157)(3,72,227,126)(4,41,228,135)(5,50,229,144)(6,59,230,153)(7,68,231,122)(8,77,232,131)(9,46,233,140)(10,55,234,149)(11,64,235,158)(12,73,236,127)(13,42,237,136)(14,51,238,145)(15,60,239,154)(16,69,240,123)(17,78,201,132)(18,47,202,141)(19,56,203,150)(20,65,204,159)(21,74,205,128)(22,43,206,137)(23,52,207,146)(24,61,208,155)(25,70,209,124)(26,79,210,133)(27,48,211,142)(28,57,212,151)(29,66,213,160)(30,75,214,129)(31,44,215,138)(32,53,216,147)(33,62,217,156)(34,71,218,125)(35,80,219,134)(36,49,220,143)(37,58,221,152)(38,67,222,121)(39,76,223,130)(40,45,224,139)(81,248,193,314)(82,257,194,283)(83,266,195,292)(84,275,196,301)(85,244,197,310)(86,253,198,319)(87,262,199,288)(88,271,200,297)(89,280,161,306)(90,249,162,315)(91,258,163,284)(92,267,164,293)(93,276,165,302)(94,245,166,311)(95,254,167,320)(96,263,168,289)(97,272,169,298)(98,241,170,307)(99,250,171,316)(100,259,172,285)(101,268,173,294)(102,277,174,303)(103,246,175,312)(104,255,176,281)(105,264,177,290)(106,273,178,299)(107,242,179,308)(108,251,180,317)(109,260,181,286)(110,269,182,295)(111,278,183,304)(112,247,184,313)(113,256,185,282)(114,265,186,291)(115,274,187,300)(116,243,188,309)(117,252,189,318)(118,261,190,287)(119,270,191,296)(120,279,192,305), (1,101,26,86,11,111,36,96,21,81,6,106,31,91,16,116)(2,110,27,95,12,120,37,105,22,90,7,115,32,100,17,85)(3,119,28,104,13,89,38,114,23,99,8,84,33,109,18,94)(4,88,29,113,14,98,39,83,24,108,9,93,34,118,19,103)(5,97,30,82,15,107,40,92,25,117,10,102,35,87,20,112)(41,297,66,282,51,307,76,292,61,317,46,302,71,287,56,312)(42,306,67,291,52,316,77,301,62,286,47,311,72,296,57,281)(43,315,68,300,53,285,78,310,63,295,48,320,73,305,58,290)(44,284,69,309,54,294,79,319,64,304,49,289,74,314,59,299)(45,293,70,318,55,303,80,288,65,313,50,298,75,283,60,308)(121,265,146,250,131,275,156,260,141,245,126,270,151,255,136,280)(122,274,147,259,132,244,157,269,142,254,127,279,152,264,137,249)(123,243,148,268,133,253,158,278,143,263,128,248,153,273,138,258)(124,252,149,277,134,262,159,247,144,272,129,257,154,242,139,267)(125,261,150,246,135,271,160,256,145,241,130,266,155,251,140,276)(161,222,186,207,171,232,196,217,181,202,166,227,191,212,176,237)(162,231,187,216,172,201,197,226,182,211,167,236,192,221,177,206)(163,240,188,225,173,210,198,235,183,220,168,205,193,230,178,215)(164,209,189,234,174,219,199,204,184,229,169,214,194,239,179,224)(165,218,190,203,175,228,200,213,185,238,170,223,195,208,180,233)>;
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), (1,54,225,148)(2,63,226,157)(3,72,227,126)(4,41,228,135)(5,50,229,144)(6,59,230,153)(7,68,231,122)(8,77,232,131)(9,46,233,140)(10,55,234,149)(11,64,235,158)(12,73,236,127)(13,42,237,136)(14,51,238,145)(15,60,239,154)(16,69,240,123)(17,78,201,132)(18,47,202,141)(19,56,203,150)(20,65,204,159)(21,74,205,128)(22,43,206,137)(23,52,207,146)(24,61,208,155)(25,70,209,124)(26,79,210,133)(27,48,211,142)(28,57,212,151)(29,66,213,160)(30,75,214,129)(31,44,215,138)(32,53,216,147)(33,62,217,156)(34,71,218,125)(35,80,219,134)(36,49,220,143)(37,58,221,152)(38,67,222,121)(39,76,223,130)(40,45,224,139)(81,248,193,314)(82,257,194,283)(83,266,195,292)(84,275,196,301)(85,244,197,310)(86,253,198,319)(87,262,199,288)(88,271,200,297)(89,280,161,306)(90,249,162,315)(91,258,163,284)(92,267,164,293)(93,276,165,302)(94,245,166,311)(95,254,167,320)(96,263,168,289)(97,272,169,298)(98,241,170,307)(99,250,171,316)(100,259,172,285)(101,268,173,294)(102,277,174,303)(103,246,175,312)(104,255,176,281)(105,264,177,290)(106,273,178,299)(107,242,179,308)(108,251,180,317)(109,260,181,286)(110,269,182,295)(111,278,183,304)(112,247,184,313)(113,256,185,282)(114,265,186,291)(115,274,187,300)(116,243,188,309)(117,252,189,318)(118,261,190,287)(119,270,191,296)(120,279,192,305), (1,101,26,86,11,111,36,96,21,81,6,106,31,91,16,116)(2,110,27,95,12,120,37,105,22,90,7,115,32,100,17,85)(3,119,28,104,13,89,38,114,23,99,8,84,33,109,18,94)(4,88,29,113,14,98,39,83,24,108,9,93,34,118,19,103)(5,97,30,82,15,107,40,92,25,117,10,102,35,87,20,112)(41,297,66,282,51,307,76,292,61,317,46,302,71,287,56,312)(42,306,67,291,52,316,77,301,62,286,47,311,72,296,57,281)(43,315,68,300,53,285,78,310,63,295,48,320,73,305,58,290)(44,284,69,309,54,294,79,319,64,304,49,289,74,314,59,299)(45,293,70,318,55,303,80,288,65,313,50,298,75,283,60,308)(121,265,146,250,131,275,156,260,141,245,126,270,151,255,136,280)(122,274,147,259,132,244,157,269,142,254,127,279,152,264,137,249)(123,243,148,268,133,253,158,278,143,263,128,248,153,273,138,258)(124,252,149,277,134,262,159,247,144,272,129,257,154,242,139,267)(125,261,150,246,135,271,160,256,145,241,130,266,155,251,140,276)(161,222,186,207,171,232,196,217,181,202,166,227,191,212,176,237)(162,231,187,216,172,201,197,226,182,211,167,236,192,221,177,206)(163,240,188,225,173,210,198,235,183,220,168,205,193,230,178,215)(164,209,189,234,174,219,199,204,184,229,169,214,194,239,179,224)(165,218,190,203,175,228,200,213,185,238,170,223,195,208,180,233) );
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)], [(1,54,225,148),(2,63,226,157),(3,72,227,126),(4,41,228,135),(5,50,229,144),(6,59,230,153),(7,68,231,122),(8,77,232,131),(9,46,233,140),(10,55,234,149),(11,64,235,158),(12,73,236,127),(13,42,237,136),(14,51,238,145),(15,60,239,154),(16,69,240,123),(17,78,201,132),(18,47,202,141),(19,56,203,150),(20,65,204,159),(21,74,205,128),(22,43,206,137),(23,52,207,146),(24,61,208,155),(25,70,209,124),(26,79,210,133),(27,48,211,142),(28,57,212,151),(29,66,213,160),(30,75,214,129),(31,44,215,138),(32,53,216,147),(33,62,217,156),(34,71,218,125),(35,80,219,134),(36,49,220,143),(37,58,221,152),(38,67,222,121),(39,76,223,130),(40,45,224,139),(81,248,193,314),(82,257,194,283),(83,266,195,292),(84,275,196,301),(85,244,197,310),(86,253,198,319),(87,262,199,288),(88,271,200,297),(89,280,161,306),(90,249,162,315),(91,258,163,284),(92,267,164,293),(93,276,165,302),(94,245,166,311),(95,254,167,320),(96,263,168,289),(97,272,169,298),(98,241,170,307),(99,250,171,316),(100,259,172,285),(101,268,173,294),(102,277,174,303),(103,246,175,312),(104,255,176,281),(105,264,177,290),(106,273,178,299),(107,242,179,308),(108,251,180,317),(109,260,181,286),(110,269,182,295),(111,278,183,304),(112,247,184,313),(113,256,185,282),(114,265,186,291),(115,274,187,300),(116,243,188,309),(117,252,189,318),(118,261,190,287),(119,270,191,296),(120,279,192,305)], [(1,101,26,86,11,111,36,96,21,81,6,106,31,91,16,116),(2,110,27,95,12,120,37,105,22,90,7,115,32,100,17,85),(3,119,28,104,13,89,38,114,23,99,8,84,33,109,18,94),(4,88,29,113,14,98,39,83,24,108,9,93,34,118,19,103),(5,97,30,82,15,107,40,92,25,117,10,102,35,87,20,112),(41,297,66,282,51,307,76,292,61,317,46,302,71,287,56,312),(42,306,67,291,52,316,77,301,62,286,47,311,72,296,57,281),(43,315,68,300,53,285,78,310,63,295,48,320,73,305,58,290),(44,284,69,309,54,294,79,319,64,304,49,289,74,314,59,299),(45,293,70,318,55,303,80,288,65,313,50,298,75,283,60,308),(121,265,146,250,131,275,156,260,141,245,126,270,151,255,136,280),(122,274,147,259,132,244,157,269,142,254,127,279,152,264,137,249),(123,243,148,268,133,253,158,278,143,263,128,248,153,273,138,258),(124,252,149,277,134,262,159,247,144,272,129,257,154,242,139,267),(125,261,150,246,135,271,160,256,145,241,130,266,155,251,140,276),(161,222,186,207,171,232,196,217,181,202,166,227,191,212,176,237),(162,231,187,216,172,201,197,226,182,211,167,236,192,221,177,206),(163,240,188,225,173,210,198,235,183,220,168,205,193,230,178,215),(164,209,189,234,174,219,199,204,184,229,169,214,194,239,179,224),(165,218,190,203,175,228,200,213,185,238,170,223,195,208,180,233)]])
104 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10F | 16A | ··· | 16H | 16I | ··· | 16P | 20A | ··· | 20H | 40A | ··· | 40P | 80A | ··· | 80AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 16 | ··· | 16 | 20 | ··· | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 2 | 2 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C16 | D4 | Q8 | D5 | M4(2) | D10 | M5(2) | Dic10 | C5⋊D4 | C4×D5 | C8⋊D5 | C8×D5 | D5×C16 | C80⋊C2 |
| kernel | C40.88D4 | C2×C5⋊2C16 | C8×Dic5 | C2×C80 | C2×C5⋊2C8 | C4×Dic5 | C2×Dic5 | Dic5 | C40 | C40 | C2×C16 | C20 | C2×C8 | C10 | C8 | C8 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 16 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 16 | 16 |
Matrix representation of C40.88D4 ►in GL3(𝔽241) generated by
| 30 | 0 | 0 |
| 0 | 189 | 51 |
| 0 | 189 | 0 |
| 240 | 0 | 0 |
| 0 | 109 | 14 |
| 0 | 12 | 132 |
| 111 | 0 | 0 |
| 0 | 157 | 98 |
| 0 | 19 | 84 |
G:=sub<GL(3,GF(241))| [30,0,0,0,189,189,0,51,0],[240,0,0,0,109,12,0,14,132],[111,0,0,0,157,19,0,98,84] >;
C40.88D4 in GAP, Magma, Sage, TeX
C_{40}._{88}D_4 % in TeX
G:=Group("C40.88D4"); // GroupNames label
G:=SmallGroup(320,59);
// by ID
G=gap.SmallGroup(320,59);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,141,36,100,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=1,c^2=a^25,b*a*b^-1=c*a*c^-1=a^9,c*b*c^-1=b^-1>;
// generators/relations
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