metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.2Q8, C10.6D16, C20.3Q16, C10.3Q32, C8.4Dic10, C5⋊2C16⋊9C4, C8.24(C4×D5), C5⋊2(C16⋊3C4), C40.48(C2×C4), (C2×C20).89D4, C2.D8.1D5, (C2×C10).31D8, C20.33(C4⋊C4), (C2×C8).219D10, C2.1(C5⋊D16), C40⋊5C4.11C2, C2.1(C5⋊Q32), C10.9(C2.D8), C4.1(C5⋊Q16), (C2×C40).71C22, C22.12(D4⋊D5), C4.2(C10.D4), C2.3(C10.D8), (C5×C2.D8).1C2, (C2×C5⋊2C16).2C2, (C2×C4).113(C5⋊D4), SmallGroup(320,47)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.2Q8
G = < a,b,c | a40=b4=1, c2=a25b2, bab-1=a31, cac-1=a9, cbc-1=a25b-1 >
Smallest permutation representation of C40.2Q8
►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)
(1 158 191 311)(2 149 192 302)(3 140 193 293)(4 131 194 284)(5 122 195 315)(6 153 196 306)(7 144 197 297)(8 135 198 288)(9 126 199 319)(10 157 200 310)(11 148 161 301)(12 139 162 292)(13 130 163 283)(14 121 164 314)(15 152 165 305)(16 143 166 296)(17 134 167 287)(18 125 168 318)(19 156 169 309)(20 147 170 300)(21 138 171 291)(22 129 172 282)(23 160 173 313)(24 151 174 304)(25 142 175 295)(26 133 176 286)(27 124 177 317)(28 155 178 308)(29 146 179 299)(30 137 180 290)(31 128 181 281)(32 159 182 312)(33 150 183 303)(34 141 184 294)(35 132 185 285)(36 123 186 316)(37 154 187 307)(38 145 188 298)(39 136 189 289)(40 127 190 320)(41 252 210 108)(42 243 211 99)(43 274 212 90)(44 265 213 81)(45 256 214 112)(46 247 215 103)(47 278 216 94)(48 269 217 85)(49 260 218 116)(50 251 219 107)(51 242 220 98)(52 273 221 89)(53 264 222 120)(54 255 223 111)(55 246 224 102)(56 277 225 93)(57 268 226 84)(58 259 227 115)(59 250 228 106)(60 241 229 97)(61 272 230 88)(62 263 231 119)(63 254 232 110)(64 245 233 101)(65 276 234 92)(66 267 235 83)(67 258 236 114)(68 249 237 105)(69 280 238 96)(70 271 239 87)(71 262 240 118)(72 253 201 109)(73 244 202 100)(74 275 203 91)(75 266 204 82)(76 257 205 113)(77 248 206 104)(78 279 207 95)(79 270 208 86)(80 261 209 117)
(1 276 176 117 11 246 186 87 21 256 196 97 31 266 166 107)(2 245 177 86 12 255 187 96 22 265 197 106 32 275 167 116)(3 254 178 95 13 264 188 105 23 274 198 115 33 244 168 85)(4 263 179 104 14 273 189 114 24 243 199 84 34 253 169 94)(5 272 180 113 15 242 190 83 25 252 200 93 35 262 170 103)(6 241 181 82 16 251 191 92 26 261 161 102 36 271 171 112)(7 250 182 91 17 260 192 101 27 270 162 111 37 280 172 81)(8 259 183 100 18 269 193 110 28 279 163 120 38 249 173 90)(9 268 184 109 19 278 194 119 29 248 164 89 39 258 174 99)(10 277 185 118 20 247 195 88 30 257 165 98 40 267 175 108)(41 295 235 127 51 305 205 137 61 315 215 147 71 285 225 157)(42 304 236 136 52 314 206 146 62 284 216 156 72 294 226 126)(43 313 237 145 53 283 207 155 63 293 217 125 73 303 227 135)(44 282 238 154 54 292 208 124 64 302 218 134 74 312 228 144)(45 291 239 123 55 301 209 133 65 311 219 143 75 281 229 153)(46 300 240 132 56 310 210 142 66 320 220 152 76 290 230 122)(47 309 201 141 57 319 211 151 67 289 221 121 77 299 231 131)(48 318 202 150 58 288 212 160 68 298 222 130 78 308 232 140)(49 287 203 159 59 297 213 129 69 307 223 139 79 317 233 149)(50 296 204 128 60 306 214 138 70 316 224 148 80 286 234 158)
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), (1,158,191,311)(2,149,192,302)(3,140,193,293)(4,131,194,284)(5,122,195,315)(6,153,196,306)(7,144,197,297)(8,135,198,288)(9,126,199,319)(10,157,200,310)(11,148,161,301)(12,139,162,292)(13,130,163,283)(14,121,164,314)(15,152,165,305)(16,143,166,296)(17,134,167,287)(18,125,168,318)(19,156,169,309)(20,147,170,300)(21,138,171,291)(22,129,172,282)(23,160,173,313)(24,151,174,304)(25,142,175,295)(26,133,176,286)(27,124,177,317)(28,155,178,308)(29,146,179,299)(30,137,180,290)(31,128,181,281)(32,159,182,312)(33,150,183,303)(34,141,184,294)(35,132,185,285)(36,123,186,316)(37,154,187,307)(38,145,188,298)(39,136,189,289)(40,127,190,320)(41,252,210,108)(42,243,211,99)(43,274,212,90)(44,265,213,81)(45,256,214,112)(46,247,215,103)(47,278,216,94)(48,269,217,85)(49,260,218,116)(50,251,219,107)(51,242,220,98)(52,273,221,89)(53,264,222,120)(54,255,223,111)(55,246,224,102)(56,277,225,93)(57,268,226,84)(58,259,227,115)(59,250,228,106)(60,241,229,97)(61,272,230,88)(62,263,231,119)(63,254,232,110)(64,245,233,101)(65,276,234,92)(66,267,235,83)(67,258,236,114)(68,249,237,105)(69,280,238,96)(70,271,239,87)(71,262,240,118)(72,253,201,109)(73,244,202,100)(74,275,203,91)(75,266,204,82)(76,257,205,113)(77,248,206,104)(78,279,207,95)(79,270,208,86)(80,261,209,117), (1,276,176,117,11,246,186,87,21,256,196,97,31,266,166,107)(2,245,177,86,12,255,187,96,22,265,197,106,32,275,167,116)(3,254,178,95,13,264,188,105,23,274,198,115,33,244,168,85)(4,263,179,104,14,273,189,114,24,243,199,84,34,253,169,94)(5,272,180,113,15,242,190,83,25,252,200,93,35,262,170,103)(6,241,181,82,16,251,191,92,26,261,161,102,36,271,171,112)(7,250,182,91,17,260,192,101,27,270,162,111,37,280,172,81)(8,259,183,100,18,269,193,110,28,279,163,120,38,249,173,90)(9,268,184,109,19,278,194,119,29,248,164,89,39,258,174,99)(10,277,185,118,20,247,195,88,30,257,165,98,40,267,175,108)(41,295,235,127,51,305,205,137,61,315,215,147,71,285,225,157)(42,304,236,136,52,314,206,146,62,284,216,156,72,294,226,126)(43,313,237,145,53,283,207,155,63,293,217,125,73,303,227,135)(44,282,238,154,54,292,208,124,64,302,218,134,74,312,228,144)(45,291,239,123,55,301,209,133,65,311,219,143,75,281,229,153)(46,300,240,132,56,310,210,142,66,320,220,152,76,290,230,122)(47,309,201,141,57,319,211,151,67,289,221,121,77,299,231,131)(48,318,202,150,58,288,212,160,68,298,222,130,78,308,232,140)(49,287,203,159,59,297,213,129,69,307,223,139,79,317,233,149)(50,296,204,128,60,306,214,138,70,316,224,148,80,286,234,158)>;
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), (1,158,191,311)(2,149,192,302)(3,140,193,293)(4,131,194,284)(5,122,195,315)(6,153,196,306)(7,144,197,297)(8,135,198,288)(9,126,199,319)(10,157,200,310)(11,148,161,301)(12,139,162,292)(13,130,163,283)(14,121,164,314)(15,152,165,305)(16,143,166,296)(17,134,167,287)(18,125,168,318)(19,156,169,309)(20,147,170,300)(21,138,171,291)(22,129,172,282)(23,160,173,313)(24,151,174,304)(25,142,175,295)(26,133,176,286)(27,124,177,317)(28,155,178,308)(29,146,179,299)(30,137,180,290)(31,128,181,281)(32,159,182,312)(33,150,183,303)(34,141,184,294)(35,132,185,285)(36,123,186,316)(37,154,187,307)(38,145,188,298)(39,136,189,289)(40,127,190,320)(41,252,210,108)(42,243,211,99)(43,274,212,90)(44,265,213,81)(45,256,214,112)(46,247,215,103)(47,278,216,94)(48,269,217,85)(49,260,218,116)(50,251,219,107)(51,242,220,98)(52,273,221,89)(53,264,222,120)(54,255,223,111)(55,246,224,102)(56,277,225,93)(57,268,226,84)(58,259,227,115)(59,250,228,106)(60,241,229,97)(61,272,230,88)(62,263,231,119)(63,254,232,110)(64,245,233,101)(65,276,234,92)(66,267,235,83)(67,258,236,114)(68,249,237,105)(69,280,238,96)(70,271,239,87)(71,262,240,118)(72,253,201,109)(73,244,202,100)(74,275,203,91)(75,266,204,82)(76,257,205,113)(77,248,206,104)(78,279,207,95)(79,270,208,86)(80,261,209,117), (1,276,176,117,11,246,186,87,21,256,196,97,31,266,166,107)(2,245,177,86,12,255,187,96,22,265,197,106,32,275,167,116)(3,254,178,95,13,264,188,105,23,274,198,115,33,244,168,85)(4,263,179,104,14,273,189,114,24,243,199,84,34,253,169,94)(5,272,180,113,15,242,190,83,25,252,200,93,35,262,170,103)(6,241,181,82,16,251,191,92,26,261,161,102,36,271,171,112)(7,250,182,91,17,260,192,101,27,270,162,111,37,280,172,81)(8,259,183,100,18,269,193,110,28,279,163,120,38,249,173,90)(9,268,184,109,19,278,194,119,29,248,164,89,39,258,174,99)(10,277,185,118,20,247,195,88,30,257,165,98,40,267,175,108)(41,295,235,127,51,305,205,137,61,315,215,147,71,285,225,157)(42,304,236,136,52,314,206,146,62,284,216,156,72,294,226,126)(43,313,237,145,53,283,207,155,63,293,217,125,73,303,227,135)(44,282,238,154,54,292,208,124,64,302,218,134,74,312,228,144)(45,291,239,123,55,301,209,133,65,311,219,143,75,281,229,153)(46,300,240,132,56,310,210,142,66,320,220,152,76,290,230,122)(47,309,201,141,57,319,211,151,67,289,221,121,77,299,231,131)(48,318,202,150,58,288,212,160,68,298,222,130,78,308,232,140)(49,287,203,159,59,297,213,129,69,307,223,139,79,317,233,149)(50,296,204,128,60,306,214,138,70,316,224,148,80,286,234,158) );
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)], [(1,158,191,311),(2,149,192,302),(3,140,193,293),(4,131,194,284),(5,122,195,315),(6,153,196,306),(7,144,197,297),(8,135,198,288),(9,126,199,319),(10,157,200,310),(11,148,161,301),(12,139,162,292),(13,130,163,283),(14,121,164,314),(15,152,165,305),(16,143,166,296),(17,134,167,287),(18,125,168,318),(19,156,169,309),(20,147,170,300),(21,138,171,291),(22,129,172,282),(23,160,173,313),(24,151,174,304),(25,142,175,295),(26,133,176,286),(27,124,177,317),(28,155,178,308),(29,146,179,299),(30,137,180,290),(31,128,181,281),(32,159,182,312),(33,150,183,303),(34,141,184,294),(35,132,185,285),(36,123,186,316),(37,154,187,307),(38,145,188,298),(39,136,189,289),(40,127,190,320),(41,252,210,108),(42,243,211,99),(43,274,212,90),(44,265,213,81),(45,256,214,112),(46,247,215,103),(47,278,216,94),(48,269,217,85),(49,260,218,116),(50,251,219,107),(51,242,220,98),(52,273,221,89),(53,264,222,120),(54,255,223,111),(55,246,224,102),(56,277,225,93),(57,268,226,84),(58,259,227,115),(59,250,228,106),(60,241,229,97),(61,272,230,88),(62,263,231,119),(63,254,232,110),(64,245,233,101),(65,276,234,92),(66,267,235,83),(67,258,236,114),(68,249,237,105),(69,280,238,96),(70,271,239,87),(71,262,240,118),(72,253,201,109),(73,244,202,100),(74,275,203,91),(75,266,204,82),(76,257,205,113),(77,248,206,104),(78,279,207,95),(79,270,208,86),(80,261,209,117)], [(1,276,176,117,11,246,186,87,21,256,196,97,31,266,166,107),(2,245,177,86,12,255,187,96,22,265,197,106,32,275,167,116),(3,254,178,95,13,264,188,105,23,274,198,115,33,244,168,85),(4,263,179,104,14,273,189,114,24,243,199,84,34,253,169,94),(5,272,180,113,15,242,190,83,25,252,200,93,35,262,170,103),(6,241,181,82,16,251,191,92,26,261,161,102,36,271,171,112),(7,250,182,91,17,260,192,101,27,270,162,111,37,280,172,81),(8,259,183,100,18,269,193,110,28,279,163,120,38,249,173,90),(9,268,184,109,19,278,194,119,29,248,164,89,39,258,174,99),(10,277,185,118,20,247,195,88,30,257,165,98,40,267,175,108),(41,295,235,127,51,305,205,137,61,315,215,147,71,285,225,157),(42,304,236,136,52,314,206,146,62,284,216,156,72,294,226,126),(43,313,237,145,53,283,207,155,63,293,217,125,73,303,227,135),(44,282,238,154,54,292,208,124,64,302,218,134,74,312,228,144),(45,291,239,123,55,301,209,133,65,311,219,143,75,281,229,153),(46,300,240,132,56,310,210,142,66,320,220,152,76,290,230,122),(47,309,201,141,57,319,211,151,67,289,221,121,77,299,231,131),(48,318,202,150,58,288,212,160,68,298,222,130,78,308,232,140),(49,287,203,159,59,297,213,129,69,307,223,139,79,317,233,149),(50,296,204,128,60,306,214,138,70,316,224,148,80,286,234,158)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 16A | ··· | 16H | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 40 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | - | + | + | + | - | - | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C4 | Q8 | D4 | D5 | Q16 | D8 | D10 | D16 | Q32 | Dic10 | C4×D5 | C5⋊D4 | C5⋊Q16 | D4⋊D5 | C5⋊D16 | C5⋊Q32 |
| kernel | C40.2Q8 | C2×C5⋊2C16 | C40⋊5C4 | C5×C2.D8 | C5⋊2C16 | C40 | C2×C20 | C2.D8 | C20 | C2×C10 | C2×C8 | C10 | C10 | C8 | C8 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 4 | 4 |
Matrix representation of C40.2Q8 ►in GL4(𝔽241) generated by
| 1 | 190 | 0 | 0 |
| 51 | 51 | 0 | 0 |
| 0 | 0 | 0 | 55 |
| 0 | 0 | 92 | 219 |
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 58 | 14 |
| 0 | 0 | 190 | 183 |
| 102 | 160 | 0 | 0 |
| 60 | 139 | 0 | 0 |
| 0 | 0 | 183 | 57 |
| 0 | 0 | 34 | 112 |
G:=sub<GL(4,GF(241))| [1,51,0,0,190,51,0,0,0,0,0,92,0,0,55,219],[64,0,0,0,0,64,0,0,0,0,58,190,0,0,14,183],[102,60,0,0,160,139,0,0,0,0,183,34,0,0,57,112] >;
C40.2Q8 in GAP, Magma, Sage, TeX
C_{40}._2Q_8 % in TeX
G:=Group("C40.2Q8"); // GroupNames label
G:=SmallGroup(320,47);
// by ID
G=gap.SmallGroup(320,47);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,141,36,346,192,851,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=1,c^2=a^25*b^2,b*a*b^-1=a^31,c*a*c^-1=a^9,c*b*c^-1=a^25*b^-1>;
// generators/relations
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