metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊4Q8, C8⋊3Dic10, C5⋊3(C8⋊Q8), C5⋊2C8⋊2Q8, C4.27(Q8×D5), C4⋊C4.44D10, (C2×C8).64D10, C2.D8.8D5, C20⋊Q8.10C2, C20.59(C2×Q8), C40⋊8C4.3C2, C2.12(C20⋊Q8), C10.17(C4⋊Q8), C40⋊6C4.10C2, (C2×Dic5).54D4, C4.24(C2×Dic10), C10.D8.9C2, C22.225(D4×D5), C20.Q8.7C2, C4.Dic10.8C2, C2.21(D8⋊D5), C10.39(C8⋊C22), (C2×C20).292C23, (C2×C40).142C22, C2.20(Q16⋊D5), C10.67(C8.C22), C4⋊Dic5.118C22, (C4×Dic5).41C22, (C5×C2.D8).7C2, (C2×C10).297(C2×D4), (C5×C4⋊C4).85C22, (C2×C5⋊2C8).66C22, (C2×C4).395(C22×D5), SmallGroup(320,503)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊4Q8
G = < a,b,c | a40=b4=1, c2=b2, bab-1=a31, cac-1=a29, cbc-1=b-1 >
Subgroups: 334 in 90 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C4, C4, C22, C5, C8, C8, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, Dic5, C20, C20, C2×C10, C8⋊C4, C4.Q8, C2.D8, C2.D8, C42.C2, C4⋊Q8, C5⋊2C8, C40, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C8⋊Q8, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C5×C4⋊C4, C2×C40, C2×Dic10, C10.D8, C20.Q8, C40⋊8C4, C40⋊6C4, C5×C2.D8, C20⋊Q8, C4.Dic10, C40⋊4Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, D10, C4⋊Q8, C8⋊C22, C8.C22, Dic10, C22×D5, C8⋊Q8, C2×Dic10, D4×D5, Q8×D5, C20⋊Q8, D8⋊D5, Q16⋊D5, C40⋊4Q8
►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)
(1 250 215 109)(2 241 216 100)(3 272 217 91)(4 263 218 82)(5 254 219 113)(6 245 220 104)(7 276 221 95)(8 267 222 86)(9 258 223 117)(10 249 224 108)(11 280 225 99)(12 271 226 90)(13 262 227 81)(14 253 228 112)(15 244 229 103)(16 275 230 94)(17 266 231 85)(18 257 232 116)(19 248 233 107)(20 279 234 98)(21 270 235 89)(22 261 236 120)(23 252 237 111)(24 243 238 102)(25 274 239 93)(26 265 240 84)(27 256 201 115)(28 247 202 106)(29 278 203 97)(30 269 204 88)(31 260 205 119)(32 251 206 110)(33 242 207 101)(34 273 208 92)(35 264 209 83)(36 255 210 114)(37 246 211 105)(38 277 212 96)(39 268 213 87)(40 259 214 118)(41 295 178 153)(42 286 179 144)(43 317 180 135)(44 308 181 126)(45 299 182 157)(46 290 183 148)(47 281 184 139)(48 312 185 130)(49 303 186 121)(50 294 187 152)(51 285 188 143)(52 316 189 134)(53 307 190 125)(54 298 191 156)(55 289 192 147)(56 320 193 138)(57 311 194 129)(58 302 195 160)(59 293 196 151)(60 284 197 142)(61 315 198 133)(62 306 199 124)(63 297 200 155)(64 288 161 146)(65 319 162 137)(66 310 163 128)(67 301 164 159)(68 292 165 150)(69 283 166 141)(70 314 167 132)(71 305 168 123)(72 296 169 154)(73 287 170 145)(74 318 171 136)(75 309 172 127)(76 300 173 158)(77 291 174 149)(78 282 175 140)(79 313 176 131)(80 304 177 122)
(1 128 215 310)(2 157 216 299)(3 146 217 288)(4 135 218 317)(5 124 219 306)(6 153 220 295)(7 142 221 284)(8 131 222 313)(9 160 223 302)(10 149 224 291)(11 138 225 320)(12 127 226 309)(13 156 227 298)(14 145 228 287)(15 134 229 316)(16 123 230 305)(17 152 231 294)(18 141 232 283)(19 130 233 312)(20 159 234 301)(21 148 235 290)(22 137 236 319)(23 126 237 308)(24 155 238 297)(25 144 239 286)(26 133 240 315)(27 122 201 304)(28 151 202 293)(29 140 203 282)(30 129 204 311)(31 158 205 300)(32 147 206 289)(33 136 207 318)(34 125 208 307)(35 154 209 296)(36 143 210 285)(37 132 211 314)(38 121 212 303)(39 150 213 292)(40 139 214 281)(41 245 178 104)(42 274 179 93)(43 263 180 82)(44 252 181 111)(45 241 182 100)(46 270 183 89)(47 259 184 118)(48 248 185 107)(49 277 186 96)(50 266 187 85)(51 255 188 114)(52 244 189 103)(53 273 190 92)(54 262 191 81)(55 251 192 110)(56 280 193 99)(57 269 194 88)(58 258 195 117)(59 247 196 106)(60 276 197 95)(61 265 198 84)(62 254 199 113)(63 243 200 102)(64 272 161 91)(65 261 162 120)(66 250 163 109)(67 279 164 98)(68 268 165 87)(69 257 166 116)(70 246 167 105)(71 275 168 94)(72 264 169 83)(73 253 170 112)(74 242 171 101)(75 271 172 90)(76 260 173 119)(77 249 174 108)(78 278 175 97)(79 267 176 86)(80 256 177 115)
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), (1,250,215,109)(2,241,216,100)(3,272,217,91)(4,263,218,82)(5,254,219,113)(6,245,220,104)(7,276,221,95)(8,267,222,86)(9,258,223,117)(10,249,224,108)(11,280,225,99)(12,271,226,90)(13,262,227,81)(14,253,228,112)(15,244,229,103)(16,275,230,94)(17,266,231,85)(18,257,232,116)(19,248,233,107)(20,279,234,98)(21,270,235,89)(22,261,236,120)(23,252,237,111)(24,243,238,102)(25,274,239,93)(26,265,240,84)(27,256,201,115)(28,247,202,106)(29,278,203,97)(30,269,204,88)(31,260,205,119)(32,251,206,110)(33,242,207,101)(34,273,208,92)(35,264,209,83)(36,255,210,114)(37,246,211,105)(38,277,212,96)(39,268,213,87)(40,259,214,118)(41,295,178,153)(42,286,179,144)(43,317,180,135)(44,308,181,126)(45,299,182,157)(46,290,183,148)(47,281,184,139)(48,312,185,130)(49,303,186,121)(50,294,187,152)(51,285,188,143)(52,316,189,134)(53,307,190,125)(54,298,191,156)(55,289,192,147)(56,320,193,138)(57,311,194,129)(58,302,195,160)(59,293,196,151)(60,284,197,142)(61,315,198,133)(62,306,199,124)(63,297,200,155)(64,288,161,146)(65,319,162,137)(66,310,163,128)(67,301,164,159)(68,292,165,150)(69,283,166,141)(70,314,167,132)(71,305,168,123)(72,296,169,154)(73,287,170,145)(74,318,171,136)(75,309,172,127)(76,300,173,158)(77,291,174,149)(78,282,175,140)(79,313,176,131)(80,304,177,122), (1,128,215,310)(2,157,216,299)(3,146,217,288)(4,135,218,317)(5,124,219,306)(6,153,220,295)(7,142,221,284)(8,131,222,313)(9,160,223,302)(10,149,224,291)(11,138,225,320)(12,127,226,309)(13,156,227,298)(14,145,228,287)(15,134,229,316)(16,123,230,305)(17,152,231,294)(18,141,232,283)(19,130,233,312)(20,159,234,301)(21,148,235,290)(22,137,236,319)(23,126,237,308)(24,155,238,297)(25,144,239,286)(26,133,240,315)(27,122,201,304)(28,151,202,293)(29,140,203,282)(30,129,204,311)(31,158,205,300)(32,147,206,289)(33,136,207,318)(34,125,208,307)(35,154,209,296)(36,143,210,285)(37,132,211,314)(38,121,212,303)(39,150,213,292)(40,139,214,281)(41,245,178,104)(42,274,179,93)(43,263,180,82)(44,252,181,111)(45,241,182,100)(46,270,183,89)(47,259,184,118)(48,248,185,107)(49,277,186,96)(50,266,187,85)(51,255,188,114)(52,244,189,103)(53,273,190,92)(54,262,191,81)(55,251,192,110)(56,280,193,99)(57,269,194,88)(58,258,195,117)(59,247,196,106)(60,276,197,95)(61,265,198,84)(62,254,199,113)(63,243,200,102)(64,272,161,91)(65,261,162,120)(66,250,163,109)(67,279,164,98)(68,268,165,87)(69,257,166,116)(70,246,167,105)(71,275,168,94)(72,264,169,83)(73,253,170,112)(74,242,171,101)(75,271,172,90)(76,260,173,119)(77,249,174,108)(78,278,175,97)(79,267,176,86)(80,256,177,115)>;
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), (1,250,215,109)(2,241,216,100)(3,272,217,91)(4,263,218,82)(5,254,219,113)(6,245,220,104)(7,276,221,95)(8,267,222,86)(9,258,223,117)(10,249,224,108)(11,280,225,99)(12,271,226,90)(13,262,227,81)(14,253,228,112)(15,244,229,103)(16,275,230,94)(17,266,231,85)(18,257,232,116)(19,248,233,107)(20,279,234,98)(21,270,235,89)(22,261,236,120)(23,252,237,111)(24,243,238,102)(25,274,239,93)(26,265,240,84)(27,256,201,115)(28,247,202,106)(29,278,203,97)(30,269,204,88)(31,260,205,119)(32,251,206,110)(33,242,207,101)(34,273,208,92)(35,264,209,83)(36,255,210,114)(37,246,211,105)(38,277,212,96)(39,268,213,87)(40,259,214,118)(41,295,178,153)(42,286,179,144)(43,317,180,135)(44,308,181,126)(45,299,182,157)(46,290,183,148)(47,281,184,139)(48,312,185,130)(49,303,186,121)(50,294,187,152)(51,285,188,143)(52,316,189,134)(53,307,190,125)(54,298,191,156)(55,289,192,147)(56,320,193,138)(57,311,194,129)(58,302,195,160)(59,293,196,151)(60,284,197,142)(61,315,198,133)(62,306,199,124)(63,297,200,155)(64,288,161,146)(65,319,162,137)(66,310,163,128)(67,301,164,159)(68,292,165,150)(69,283,166,141)(70,314,167,132)(71,305,168,123)(72,296,169,154)(73,287,170,145)(74,318,171,136)(75,309,172,127)(76,300,173,158)(77,291,174,149)(78,282,175,140)(79,313,176,131)(80,304,177,122), (1,128,215,310)(2,157,216,299)(3,146,217,288)(4,135,218,317)(5,124,219,306)(6,153,220,295)(7,142,221,284)(8,131,222,313)(9,160,223,302)(10,149,224,291)(11,138,225,320)(12,127,226,309)(13,156,227,298)(14,145,228,287)(15,134,229,316)(16,123,230,305)(17,152,231,294)(18,141,232,283)(19,130,233,312)(20,159,234,301)(21,148,235,290)(22,137,236,319)(23,126,237,308)(24,155,238,297)(25,144,239,286)(26,133,240,315)(27,122,201,304)(28,151,202,293)(29,140,203,282)(30,129,204,311)(31,158,205,300)(32,147,206,289)(33,136,207,318)(34,125,208,307)(35,154,209,296)(36,143,210,285)(37,132,211,314)(38,121,212,303)(39,150,213,292)(40,139,214,281)(41,245,178,104)(42,274,179,93)(43,263,180,82)(44,252,181,111)(45,241,182,100)(46,270,183,89)(47,259,184,118)(48,248,185,107)(49,277,186,96)(50,266,187,85)(51,255,188,114)(52,244,189,103)(53,273,190,92)(54,262,191,81)(55,251,192,110)(56,280,193,99)(57,269,194,88)(58,258,195,117)(59,247,196,106)(60,276,197,95)(61,265,198,84)(62,254,199,113)(63,243,200,102)(64,272,161,91)(65,261,162,120)(66,250,163,109)(67,279,164,98)(68,268,165,87)(69,257,166,116)(70,246,167,105)(71,275,168,94)(72,264,169,83)(73,253,170,112)(74,242,171,101)(75,271,172,90)(76,260,173,119)(77,249,174,108)(78,278,175,97)(79,267,176,86)(80,256,177,115) );
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)], [(1,250,215,109),(2,241,216,100),(3,272,217,91),(4,263,218,82),(5,254,219,113),(6,245,220,104),(7,276,221,95),(8,267,222,86),(9,258,223,117),(10,249,224,108),(11,280,225,99),(12,271,226,90),(13,262,227,81),(14,253,228,112),(15,244,229,103),(16,275,230,94),(17,266,231,85),(18,257,232,116),(19,248,233,107),(20,279,234,98),(21,270,235,89),(22,261,236,120),(23,252,237,111),(24,243,238,102),(25,274,239,93),(26,265,240,84),(27,256,201,115),(28,247,202,106),(29,278,203,97),(30,269,204,88),(31,260,205,119),(32,251,206,110),(33,242,207,101),(34,273,208,92),(35,264,209,83),(36,255,210,114),(37,246,211,105),(38,277,212,96),(39,268,213,87),(40,259,214,118),(41,295,178,153),(42,286,179,144),(43,317,180,135),(44,308,181,126),(45,299,182,157),(46,290,183,148),(47,281,184,139),(48,312,185,130),(49,303,186,121),(50,294,187,152),(51,285,188,143),(52,316,189,134),(53,307,190,125),(54,298,191,156),(55,289,192,147),(56,320,193,138),(57,311,194,129),(58,302,195,160),(59,293,196,151),(60,284,197,142),(61,315,198,133),(62,306,199,124),(63,297,200,155),(64,288,161,146),(65,319,162,137),(66,310,163,128),(67,301,164,159),(68,292,165,150),(69,283,166,141),(70,314,167,132),(71,305,168,123),(72,296,169,154),(73,287,170,145),(74,318,171,136),(75,309,172,127),(76,300,173,158),(77,291,174,149),(78,282,175,140),(79,313,176,131),(80,304,177,122)], [(1,128,215,310),(2,157,216,299),(3,146,217,288),(4,135,218,317),(5,124,219,306),(6,153,220,295),(7,142,221,284),(8,131,222,313),(9,160,223,302),(10,149,224,291),(11,138,225,320),(12,127,226,309),(13,156,227,298),(14,145,228,287),(15,134,229,316),(16,123,230,305),(17,152,231,294),(18,141,232,283),(19,130,233,312),(20,159,234,301),(21,148,235,290),(22,137,236,319),(23,126,237,308),(24,155,238,297),(25,144,239,286),(26,133,240,315),(27,122,201,304),(28,151,202,293),(29,140,203,282),(30,129,204,311),(31,158,205,300),(32,147,206,289),(33,136,207,318),(34,125,208,307),(35,154,209,296),(36,143,210,285),(37,132,211,314),(38,121,212,303),(39,150,213,292),(40,139,214,281),(41,245,178,104),(42,274,179,93),(43,263,180,82),(44,252,181,111),(45,241,182,100),(46,270,183,89),(47,259,184,118),(48,248,185,107),(49,277,186,96),(50,266,187,85),(51,255,188,114),(52,244,189,103),(53,273,190,92),(54,262,191,81),(55,251,192,110),(56,280,193,99),(57,269,194,88),(58,258,195,117),(59,247,196,106),(60,276,197,95),(61,265,198,84),(62,254,199,113),(63,243,200,102),(64,272,161,91),(65,261,162,120),(66,250,163,109),(67,279,164,98),(68,268,165,87),(69,257,166,116),(70,246,167,105),(71,275,168,94),(72,264,169,83),(73,253,170,112),(74,242,171,101),(75,271,172,90),(76,260,173,119),(77,249,174,108),(78,278,175,97),(79,267,176,86),(80,256,177,115)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 20 | 20 | 40 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | + | + | + | + | - | + | - | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | Q8 | D4 | D5 | D10 | D10 | Dic10 | C8⋊C22 | C8.C22 | Q8×D5 | D4×D5 | D8⋊D5 | Q16⋊D5 |
| kernel | C40⋊4Q8 | C10.D8 | C20.Q8 | C40⋊8C4 | C40⋊6C4 | C5×C2.D8 | C20⋊Q8 | C4.Dic10 | C5⋊2C8 | C40 | C2×Dic5 | C2.D8 | C4⋊C4 | C2×C8 | C8 | C10 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 8 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C40⋊4Q8 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 6 | 11 | 6 |
| 0 | 0 | 35 | 6 | 35 | 6 |
| 0 | 0 | 30 | 35 | 11 | 6 |
| 0 | 0 | 6 | 35 | 35 | 6 |
| 25 | 9 | 0 | 0 | 0 | 0 |
| 17 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 15 | 28 | 3 |
| 0 | 0 | 26 | 13 | 38 | 5 |
| 0 | 0 | 28 | 3 | 36 | 26 |
| 0 | 0 | 38 | 5 | 15 | 28 |
| 25 | 22 | 0 | 0 | 0 | 0 |
| 20 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 35 | 21 | 15 |
| 0 | 0 | 3 | 19 | 18 | 20 |
| 0 | 0 | 20 | 26 | 22 | 35 |
| 0 | 0 | 23 | 21 | 3 | 19 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,35,30,6,0,0,6,6,35,35,0,0,11,35,11,35,0,0,6,6,6,6],[25,17,0,0,0,0,9,16,0,0,0,0,0,0,5,26,28,38,0,0,15,13,3,5,0,0,28,38,36,15,0,0,3,5,26,28],[25,20,0,0,0,0,22,16,0,0,0,0,0,0,22,3,20,23,0,0,35,19,26,21,0,0,21,18,22,3,0,0,15,20,35,19] >;
C40⋊4Q8 in GAP, Magma, Sage, TeX
C_{40}\rtimes_4Q_8 % in TeX
G:=Group("C40:4Q8"); // GroupNames label
G:=SmallGroup(320,503);
// by ID
G=gap.SmallGroup(320,503);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,120,254,219,58,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=1,c^2=b^2,b*a*b^-1=a^31,c*a*c^-1=a^29,c*b*c^-1=b^-1>;
// generators/relations