metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊8Q8, C4.4D40, C8⋊7Dic10, C20.29D8, C20.15Q16, C4.4Dic20, C42.254D10, (C4×C8).7D5, (C4×C40).9C2, C10.1(C2×D8), C5⋊1(C8⋊2Q8), C2.4(C2×D40), (C2×C4).78D20, C10.3(C4⋊Q8), C10.2(C2×Q16), C20.70(C2×Q8), C40⋊5C4.4C2, (C2×C8).300D10, (C2×C20).375D4, C20⋊2Q8.4C2, C2.5(C2×Dic20), C2.7(C20⋊2Q8), C4.36(C2×Dic10), C22.87(C2×D20), C4⋊Dic5.3C22, (C2×C20).718C23, (C2×C40).373C22, (C4×C20).304C22, (C2×C10).101(C2×D4), (C2×C4).661(C22×D5), SmallGroup(320,309)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊8Q8
G = < a,b,c | a40=b4=1, c2=b2, ab=ba, cac-1=a-1, cbc-1=b-1 >
Subgroups: 398 in 98 conjugacy classes, 55 normal (21 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C2×C8, C2×Q8, Dic5, C20, C2×C10, C4×C8, C2.D8, C4⋊Q8, C40, Dic10, C2×Dic5, C2×C20, C8⋊2Q8, C4⋊Dic5, C4⋊Dic5, C4×C20, C2×C40, C2×Dic10, C40⋊5C4, C4×C40, C20⋊2Q8, C40⋊8Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, D8, Q16, C2×D4, C2×Q8, D10, C4⋊Q8, C2×D8, C2×Q16, Dic10, D20, C22×D5, C8⋊2Q8, D40, Dic20, C2×Dic10, C2×D20, C20⋊2Q8, C2×D40, C2×Dic20, C40⋊8Q8
►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)
(1 205 199 82)(2 206 200 83)(3 207 161 84)(4 208 162 85)(5 209 163 86)(6 210 164 87)(7 211 165 88)(8 212 166 89)(9 213 167 90)(10 214 168 91)(11 215 169 92)(12 216 170 93)(13 217 171 94)(14 218 172 95)(15 219 173 96)(16 220 174 97)(17 221 175 98)(18 222 176 99)(19 223 177 100)(20 224 178 101)(21 225 179 102)(22 226 180 103)(23 227 181 104)(24 228 182 105)(25 229 183 106)(26 230 184 107)(27 231 185 108)(28 232 186 109)(29 233 187 110)(30 234 188 111)(31 235 189 112)(32 236 190 113)(33 237 191 114)(34 238 192 115)(35 239 193 116)(36 240 194 117)(37 201 195 118)(38 202 196 119)(39 203 197 120)(40 204 198 81)(41 152 248 309)(42 153 249 310)(43 154 250 311)(44 155 251 312)(45 156 252 313)(46 157 253 314)(47 158 254 315)(48 159 255 316)(49 160 256 317)(50 121 257 318)(51 122 258 319)(52 123 259 320)(53 124 260 281)(54 125 261 282)(55 126 262 283)(56 127 263 284)(57 128 264 285)(58 129 265 286)(59 130 266 287)(60 131 267 288)(61 132 268 289)(62 133 269 290)(63 134 270 291)(64 135 271 292)(65 136 272 293)(66 137 273 294)(67 138 274 295)(68 139 275 296)(69 140 276 297)(70 141 277 298)(71 142 278 299)(72 143 279 300)(73 144 280 301)(74 145 241 302)(75 146 242 303)(76 147 243 304)(77 148 244 305)(78 149 245 306)(79 150 246 307)(80 151 247 308)
(1 305 199 148)(2 304 200 147)(3 303 161 146)(4 302 162 145)(5 301 163 144)(6 300 164 143)(7 299 165 142)(8 298 166 141)(9 297 167 140)(10 296 168 139)(11 295 169 138)(12 294 170 137)(13 293 171 136)(14 292 172 135)(15 291 173 134)(16 290 174 133)(17 289 175 132)(18 288 176 131)(19 287 177 130)(20 286 178 129)(21 285 179 128)(22 284 180 127)(23 283 181 126)(24 282 182 125)(25 281 183 124)(26 320 184 123)(27 319 185 122)(28 318 186 121)(29 317 187 160)(30 316 188 159)(31 315 189 158)(32 314 190 157)(33 313 191 156)(34 312 192 155)(35 311 193 154)(36 310 194 153)(37 309 195 152)(38 308 196 151)(39 307 197 150)(40 306 198 149)(41 201 248 118)(42 240 249 117)(43 239 250 116)(44 238 251 115)(45 237 252 114)(46 236 253 113)(47 235 254 112)(48 234 255 111)(49 233 256 110)(50 232 257 109)(51 231 258 108)(52 230 259 107)(53 229 260 106)(54 228 261 105)(55 227 262 104)(56 226 263 103)(57 225 264 102)(58 224 265 101)(59 223 266 100)(60 222 267 99)(61 221 268 98)(62 220 269 97)(63 219 270 96)(64 218 271 95)(65 217 272 94)(66 216 273 93)(67 215 274 92)(68 214 275 91)(69 213 276 90)(70 212 277 89)(71 211 278 88)(72 210 279 87)(73 209 280 86)(74 208 241 85)(75 207 242 84)(76 206 243 83)(77 205 244 82)(78 204 245 81)(79 203 246 120)(80 202 247 119)
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), (1,205,199,82)(2,206,200,83)(3,207,161,84)(4,208,162,85)(5,209,163,86)(6,210,164,87)(7,211,165,88)(8,212,166,89)(9,213,167,90)(10,214,168,91)(11,215,169,92)(12,216,170,93)(13,217,171,94)(14,218,172,95)(15,219,173,96)(16,220,174,97)(17,221,175,98)(18,222,176,99)(19,223,177,100)(20,224,178,101)(21,225,179,102)(22,226,180,103)(23,227,181,104)(24,228,182,105)(25,229,183,106)(26,230,184,107)(27,231,185,108)(28,232,186,109)(29,233,187,110)(30,234,188,111)(31,235,189,112)(32,236,190,113)(33,237,191,114)(34,238,192,115)(35,239,193,116)(36,240,194,117)(37,201,195,118)(38,202,196,119)(39,203,197,120)(40,204,198,81)(41,152,248,309)(42,153,249,310)(43,154,250,311)(44,155,251,312)(45,156,252,313)(46,157,253,314)(47,158,254,315)(48,159,255,316)(49,160,256,317)(50,121,257,318)(51,122,258,319)(52,123,259,320)(53,124,260,281)(54,125,261,282)(55,126,262,283)(56,127,263,284)(57,128,264,285)(58,129,265,286)(59,130,266,287)(60,131,267,288)(61,132,268,289)(62,133,269,290)(63,134,270,291)(64,135,271,292)(65,136,272,293)(66,137,273,294)(67,138,274,295)(68,139,275,296)(69,140,276,297)(70,141,277,298)(71,142,278,299)(72,143,279,300)(73,144,280,301)(74,145,241,302)(75,146,242,303)(76,147,243,304)(77,148,244,305)(78,149,245,306)(79,150,246,307)(80,151,247,308), (1,305,199,148)(2,304,200,147)(3,303,161,146)(4,302,162,145)(5,301,163,144)(6,300,164,143)(7,299,165,142)(8,298,166,141)(9,297,167,140)(10,296,168,139)(11,295,169,138)(12,294,170,137)(13,293,171,136)(14,292,172,135)(15,291,173,134)(16,290,174,133)(17,289,175,132)(18,288,176,131)(19,287,177,130)(20,286,178,129)(21,285,179,128)(22,284,180,127)(23,283,181,126)(24,282,182,125)(25,281,183,124)(26,320,184,123)(27,319,185,122)(28,318,186,121)(29,317,187,160)(30,316,188,159)(31,315,189,158)(32,314,190,157)(33,313,191,156)(34,312,192,155)(35,311,193,154)(36,310,194,153)(37,309,195,152)(38,308,196,151)(39,307,197,150)(40,306,198,149)(41,201,248,118)(42,240,249,117)(43,239,250,116)(44,238,251,115)(45,237,252,114)(46,236,253,113)(47,235,254,112)(48,234,255,111)(49,233,256,110)(50,232,257,109)(51,231,258,108)(52,230,259,107)(53,229,260,106)(54,228,261,105)(55,227,262,104)(56,226,263,103)(57,225,264,102)(58,224,265,101)(59,223,266,100)(60,222,267,99)(61,221,268,98)(62,220,269,97)(63,219,270,96)(64,218,271,95)(65,217,272,94)(66,216,273,93)(67,215,274,92)(68,214,275,91)(69,213,276,90)(70,212,277,89)(71,211,278,88)(72,210,279,87)(73,209,280,86)(74,208,241,85)(75,207,242,84)(76,206,243,83)(77,205,244,82)(78,204,245,81)(79,203,246,120)(80,202,247,119)>;
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), (1,205,199,82)(2,206,200,83)(3,207,161,84)(4,208,162,85)(5,209,163,86)(6,210,164,87)(7,211,165,88)(8,212,166,89)(9,213,167,90)(10,214,168,91)(11,215,169,92)(12,216,170,93)(13,217,171,94)(14,218,172,95)(15,219,173,96)(16,220,174,97)(17,221,175,98)(18,222,176,99)(19,223,177,100)(20,224,178,101)(21,225,179,102)(22,226,180,103)(23,227,181,104)(24,228,182,105)(25,229,183,106)(26,230,184,107)(27,231,185,108)(28,232,186,109)(29,233,187,110)(30,234,188,111)(31,235,189,112)(32,236,190,113)(33,237,191,114)(34,238,192,115)(35,239,193,116)(36,240,194,117)(37,201,195,118)(38,202,196,119)(39,203,197,120)(40,204,198,81)(41,152,248,309)(42,153,249,310)(43,154,250,311)(44,155,251,312)(45,156,252,313)(46,157,253,314)(47,158,254,315)(48,159,255,316)(49,160,256,317)(50,121,257,318)(51,122,258,319)(52,123,259,320)(53,124,260,281)(54,125,261,282)(55,126,262,283)(56,127,263,284)(57,128,264,285)(58,129,265,286)(59,130,266,287)(60,131,267,288)(61,132,268,289)(62,133,269,290)(63,134,270,291)(64,135,271,292)(65,136,272,293)(66,137,273,294)(67,138,274,295)(68,139,275,296)(69,140,276,297)(70,141,277,298)(71,142,278,299)(72,143,279,300)(73,144,280,301)(74,145,241,302)(75,146,242,303)(76,147,243,304)(77,148,244,305)(78,149,245,306)(79,150,246,307)(80,151,247,308), (1,305,199,148)(2,304,200,147)(3,303,161,146)(4,302,162,145)(5,301,163,144)(6,300,164,143)(7,299,165,142)(8,298,166,141)(9,297,167,140)(10,296,168,139)(11,295,169,138)(12,294,170,137)(13,293,171,136)(14,292,172,135)(15,291,173,134)(16,290,174,133)(17,289,175,132)(18,288,176,131)(19,287,177,130)(20,286,178,129)(21,285,179,128)(22,284,180,127)(23,283,181,126)(24,282,182,125)(25,281,183,124)(26,320,184,123)(27,319,185,122)(28,318,186,121)(29,317,187,160)(30,316,188,159)(31,315,189,158)(32,314,190,157)(33,313,191,156)(34,312,192,155)(35,311,193,154)(36,310,194,153)(37,309,195,152)(38,308,196,151)(39,307,197,150)(40,306,198,149)(41,201,248,118)(42,240,249,117)(43,239,250,116)(44,238,251,115)(45,237,252,114)(46,236,253,113)(47,235,254,112)(48,234,255,111)(49,233,256,110)(50,232,257,109)(51,231,258,108)(52,230,259,107)(53,229,260,106)(54,228,261,105)(55,227,262,104)(56,226,263,103)(57,225,264,102)(58,224,265,101)(59,223,266,100)(60,222,267,99)(61,221,268,98)(62,220,269,97)(63,219,270,96)(64,218,271,95)(65,217,272,94)(66,216,273,93)(67,215,274,92)(68,214,275,91)(69,213,276,90)(70,212,277,89)(71,211,278,88)(72,210,279,87)(73,209,280,86)(74,208,241,85)(75,207,242,84)(76,206,243,83)(77,205,244,82)(78,204,245,81)(79,203,246,120)(80,202,247,119) );
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)], [(1,205,199,82),(2,206,200,83),(3,207,161,84),(4,208,162,85),(5,209,163,86),(6,210,164,87),(7,211,165,88),(8,212,166,89),(9,213,167,90),(10,214,168,91),(11,215,169,92),(12,216,170,93),(13,217,171,94),(14,218,172,95),(15,219,173,96),(16,220,174,97),(17,221,175,98),(18,222,176,99),(19,223,177,100),(20,224,178,101),(21,225,179,102),(22,226,180,103),(23,227,181,104),(24,228,182,105),(25,229,183,106),(26,230,184,107),(27,231,185,108),(28,232,186,109),(29,233,187,110),(30,234,188,111),(31,235,189,112),(32,236,190,113),(33,237,191,114),(34,238,192,115),(35,239,193,116),(36,240,194,117),(37,201,195,118),(38,202,196,119),(39,203,197,120),(40,204,198,81),(41,152,248,309),(42,153,249,310),(43,154,250,311),(44,155,251,312),(45,156,252,313),(46,157,253,314),(47,158,254,315),(48,159,255,316),(49,160,256,317),(50,121,257,318),(51,122,258,319),(52,123,259,320),(53,124,260,281),(54,125,261,282),(55,126,262,283),(56,127,263,284),(57,128,264,285),(58,129,265,286),(59,130,266,287),(60,131,267,288),(61,132,268,289),(62,133,269,290),(63,134,270,291),(64,135,271,292),(65,136,272,293),(66,137,273,294),(67,138,274,295),(68,139,275,296),(69,140,276,297),(70,141,277,298),(71,142,278,299),(72,143,279,300),(73,144,280,301),(74,145,241,302),(75,146,242,303),(76,147,243,304),(77,148,244,305),(78,149,245,306),(79,150,246,307),(80,151,247,308)], [(1,305,199,148),(2,304,200,147),(3,303,161,146),(4,302,162,145),(5,301,163,144),(6,300,164,143),(7,299,165,142),(8,298,166,141),(9,297,167,140),(10,296,168,139),(11,295,169,138),(12,294,170,137),(13,293,171,136),(14,292,172,135),(15,291,173,134),(16,290,174,133),(17,289,175,132),(18,288,176,131),(19,287,177,130),(20,286,178,129),(21,285,179,128),(22,284,180,127),(23,283,181,126),(24,282,182,125),(25,281,183,124),(26,320,184,123),(27,319,185,122),(28,318,186,121),(29,317,187,160),(30,316,188,159),(31,315,189,158),(32,314,190,157),(33,313,191,156),(34,312,192,155),(35,311,193,154),(36,310,194,153),(37,309,195,152),(38,308,196,151),(39,307,197,150),(40,306,198,149),(41,201,248,118),(42,240,249,117),(43,239,250,116),(44,238,251,115),(45,237,252,114),(46,236,253,113),(47,235,254,112),(48,234,255,111),(49,233,256,110),(50,232,257,109),(51,231,258,108),(52,230,259,107),(53,229,260,106),(54,228,261,105),(55,227,262,104),(56,226,263,103),(57,225,264,102),(58,224,265,101),(59,223,266,100),(60,222,267,99),(61,221,268,98),(62,220,269,97),(63,219,270,96),(64,218,271,95),(65,217,272,94),(66,216,273,93),(67,215,274,92),(68,214,275,91),(69,213,276,90),(70,212,277,89),(71,211,278,88),(72,210,279,87),(73,209,280,86),(74,208,241,85),(75,207,242,84),(76,206,243,83),(77,205,244,82),(78,204,245,81),(79,203,246,120),(80,202,247,119)]])
86 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 20A | ··· | 20X | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 40 | 40 | 40 | 40 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
86 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | + | - | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | Q8 | D4 | D5 | D8 | Q16 | D10 | D10 | Dic10 | D20 | D40 | Dic20 |
| kernel | C40⋊8Q8 | C40⋊5C4 | C4×C40 | C20⋊2Q8 | C40 | C2×C20 | C4×C8 | C20 | C20 | C42 | C2×C8 | C8 | C2×C4 | C4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 4 | 2 | 2 | 4 | 4 | 2 | 4 | 16 | 8 | 16 | 16 |
Matrix representation of C40⋊8Q8 ►in GL4(𝔽41) generated by
| 0 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 22 | 0 |
| 0 | 0 | 28 | 28 |
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 2 | 9 |
| 1 | 30 | 0 | 0 |
| 30 | 40 | 0 | 0 |
| 0 | 0 | 18 | 39 |
| 0 | 0 | 19 | 23 |
G:=sub<GL(4,GF(41))| [0,1,0,0,40,0,0,0,0,0,22,28,0,0,0,28],[0,40,0,0,1,0,0,0,0,0,32,2,0,0,0,9],[1,30,0,0,30,40,0,0,0,0,18,19,0,0,39,23] >;
C40⋊8Q8 in GAP, Magma, Sage, TeX
C_{40}\rtimes_8Q_8 % in TeX
G:=Group("C40:8Q8"); // GroupNames label
G:=SmallGroup(320,309);
// by ID
G=gap.SmallGroup(320,309);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,120,254,226,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations