metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊9Q8, C8⋊8Dic10, C20.24SD16, C42.252D10, C5⋊1(C8⋊3Q8), (C4×C8).13D5, (C4×C40).15C2, (C2×C4).76D20, C10.2(C4⋊Q8), C20.69(C2×Q8), C40⋊6C4.4C2, (C2×C8).315D10, (C2×C20).373D4, C4.3(C40⋊C2), C20⋊2Q8.2C2, C10.1(C2×SD16), C2.6(C20⋊2Q8), C4.35(C2×Dic10), C22.85(C2×D20), C4⋊Dic5.1C22, (C4×C20).302C22, (C2×C40).387C22, (C2×C20).716C23, C2.5(C2×C40⋊C2), (C2×C10).99(C2×D4), (C2×C4).659(C22×D5), SmallGroup(320,307)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊9Q8
G = < a,b,c | a40=b4=1, c2=b2, ab=ba, cac-1=a19, cbc-1=b-1 >
Subgroups: 398 in 98 conjugacy classes, 55 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, C2×C4, Q8, C10, C10, C42, C4⋊C4, C2×C8, C2×Q8, Dic5, C20, C2×C10, C4×C8, C4.Q8, C4⋊Q8, C40, Dic10, C2×Dic5, C2×C20, C2×C20, C8⋊3Q8, C4⋊Dic5, C4⋊Dic5, C4×C20, C2×C40, C2×Dic10, C40⋊6C4, C4×C40, C20⋊2Q8, C40⋊9Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, SD16, C2×D4, C2×Q8, D10, C4⋊Q8, C2×SD16, Dic10, D20, C22×D5, C8⋊3Q8, C40⋊C2, C2×Dic10, C2×D20, C20⋊2Q8, C2×C40⋊C2, C40⋊9Q8
►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 75 201 246)(2 76 202 247)(3 77 203 248)(4 78 204 249)(5 79 205 250)(6 80 206 251)(7 41 207 252)(8 42 208 253)(9 43 209 254)(10 44 210 255)(11 45 211 256)(12 46 212 257)(13 47 213 258)(14 48 214 259)(15 49 215 260)(16 50 216 261)(17 51 217 262)(18 52 218 263)(19 53 219 264)(20 54 220 265)(21 55 221 266)(22 56 222 267)(23 57 223 268)(24 58 224 269)(25 59 225 270)(26 60 226 271)(27 61 227 272)(28 62 228 273)(29 63 229 274)(30 64 230 275)(31 65 231 276)(32 66 232 277)(33 67 233 278)(34 68 234 279)(35 69 235 280)(36 70 236 241)(37 71 237 242)(38 72 238 243)(39 73 239 244)(40 74 240 245)(81 140 199 318)(82 141 200 319)(83 142 161 320)(84 143 162 281)(85 144 163 282)(86 145 164 283)(87 146 165 284)(88 147 166 285)(89 148 167 286)(90 149 168 287)(91 150 169 288)(92 151 170 289)(93 152 171 290)(94 153 172 291)(95 154 173 292)(96 155 174 293)(97 156 175 294)(98 157 176 295)(99 158 177 296)(100 159 178 297)(101 160 179 298)(102 121 180 299)(103 122 181 300)(104 123 182 301)(105 124 183 302)(106 125 184 303)(107 126 185 304)(108 127 186 305)(109 128 187 306)(110 129 188 307)(111 130 189 308)(112 131 190 309)(113 132 191 310)(114 133 192 311)(115 134 193 312)(116 135 194 313)(117 136 195 314)(118 137 196 315)(119 138 197 316)(120 139 198 317)
(1 115 201 193)(2 94 202 172)(3 113 203 191)(4 92 204 170)(5 111 205 189)(6 90 206 168)(7 109 207 187)(8 88 208 166)(9 107 209 185)(10 86 210 164)(11 105 211 183)(12 84 212 162)(13 103 213 181)(14 82 214 200)(15 101 215 179)(16 120 216 198)(17 99 217 177)(18 118 218 196)(19 97 219 175)(20 116 220 194)(21 95 221 173)(22 114 222 192)(23 93 223 171)(24 112 224 190)(25 91 225 169)(26 110 226 188)(27 89 227 167)(28 108 228 186)(29 87 229 165)(30 106 230 184)(31 85 231 163)(32 104 232 182)(33 83 233 161)(34 102 234 180)(35 81 235 199)(36 100 236 178)(37 119 237 197)(38 98 238 176)(39 117 239 195)(40 96 240 174)(41 306 252 128)(42 285 253 147)(43 304 254 126)(44 283 255 145)(45 302 256 124)(46 281 257 143)(47 300 258 122)(48 319 259 141)(49 298 260 160)(50 317 261 139)(51 296 262 158)(52 315 263 137)(53 294 264 156)(54 313 265 135)(55 292 266 154)(56 311 267 133)(57 290 268 152)(58 309 269 131)(59 288 270 150)(60 307 271 129)(61 286 272 148)(62 305 273 127)(63 284 274 146)(64 303 275 125)(65 282 276 144)(66 301 277 123)(67 320 278 142)(68 299 279 121)(69 318 280 140)(70 297 241 159)(71 316 242 138)(72 295 243 157)(73 314 244 136)(74 293 245 155)(75 312 246 134)(76 291 247 153)(77 310 248 132)(78 289 249 151)(79 308 250 130)(80 287 251 149)
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), (1,75,201,246)(2,76,202,247)(3,77,203,248)(4,78,204,249)(5,79,205,250)(6,80,206,251)(7,41,207,252)(8,42,208,253)(9,43,209,254)(10,44,210,255)(11,45,211,256)(12,46,212,257)(13,47,213,258)(14,48,214,259)(15,49,215,260)(16,50,216,261)(17,51,217,262)(18,52,218,263)(19,53,219,264)(20,54,220,265)(21,55,221,266)(22,56,222,267)(23,57,223,268)(24,58,224,269)(25,59,225,270)(26,60,226,271)(27,61,227,272)(28,62,228,273)(29,63,229,274)(30,64,230,275)(31,65,231,276)(32,66,232,277)(33,67,233,278)(34,68,234,279)(35,69,235,280)(36,70,236,241)(37,71,237,242)(38,72,238,243)(39,73,239,244)(40,74,240,245)(81,140,199,318)(82,141,200,319)(83,142,161,320)(84,143,162,281)(85,144,163,282)(86,145,164,283)(87,146,165,284)(88,147,166,285)(89,148,167,286)(90,149,168,287)(91,150,169,288)(92,151,170,289)(93,152,171,290)(94,153,172,291)(95,154,173,292)(96,155,174,293)(97,156,175,294)(98,157,176,295)(99,158,177,296)(100,159,178,297)(101,160,179,298)(102,121,180,299)(103,122,181,300)(104,123,182,301)(105,124,183,302)(106,125,184,303)(107,126,185,304)(108,127,186,305)(109,128,187,306)(110,129,188,307)(111,130,189,308)(112,131,190,309)(113,132,191,310)(114,133,192,311)(115,134,193,312)(116,135,194,313)(117,136,195,314)(118,137,196,315)(119,138,197,316)(120,139,198,317), (1,115,201,193)(2,94,202,172)(3,113,203,191)(4,92,204,170)(5,111,205,189)(6,90,206,168)(7,109,207,187)(8,88,208,166)(9,107,209,185)(10,86,210,164)(11,105,211,183)(12,84,212,162)(13,103,213,181)(14,82,214,200)(15,101,215,179)(16,120,216,198)(17,99,217,177)(18,118,218,196)(19,97,219,175)(20,116,220,194)(21,95,221,173)(22,114,222,192)(23,93,223,171)(24,112,224,190)(25,91,225,169)(26,110,226,188)(27,89,227,167)(28,108,228,186)(29,87,229,165)(30,106,230,184)(31,85,231,163)(32,104,232,182)(33,83,233,161)(34,102,234,180)(35,81,235,199)(36,100,236,178)(37,119,237,197)(38,98,238,176)(39,117,239,195)(40,96,240,174)(41,306,252,128)(42,285,253,147)(43,304,254,126)(44,283,255,145)(45,302,256,124)(46,281,257,143)(47,300,258,122)(48,319,259,141)(49,298,260,160)(50,317,261,139)(51,296,262,158)(52,315,263,137)(53,294,264,156)(54,313,265,135)(55,292,266,154)(56,311,267,133)(57,290,268,152)(58,309,269,131)(59,288,270,150)(60,307,271,129)(61,286,272,148)(62,305,273,127)(63,284,274,146)(64,303,275,125)(65,282,276,144)(66,301,277,123)(67,320,278,142)(68,299,279,121)(69,318,280,140)(70,297,241,159)(71,316,242,138)(72,295,243,157)(73,314,244,136)(74,293,245,155)(75,312,246,134)(76,291,247,153)(77,310,248,132)(78,289,249,151)(79,308,250,130)(80,287,251,149)>;
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), (1,75,201,246)(2,76,202,247)(3,77,203,248)(4,78,204,249)(5,79,205,250)(6,80,206,251)(7,41,207,252)(8,42,208,253)(9,43,209,254)(10,44,210,255)(11,45,211,256)(12,46,212,257)(13,47,213,258)(14,48,214,259)(15,49,215,260)(16,50,216,261)(17,51,217,262)(18,52,218,263)(19,53,219,264)(20,54,220,265)(21,55,221,266)(22,56,222,267)(23,57,223,268)(24,58,224,269)(25,59,225,270)(26,60,226,271)(27,61,227,272)(28,62,228,273)(29,63,229,274)(30,64,230,275)(31,65,231,276)(32,66,232,277)(33,67,233,278)(34,68,234,279)(35,69,235,280)(36,70,236,241)(37,71,237,242)(38,72,238,243)(39,73,239,244)(40,74,240,245)(81,140,199,318)(82,141,200,319)(83,142,161,320)(84,143,162,281)(85,144,163,282)(86,145,164,283)(87,146,165,284)(88,147,166,285)(89,148,167,286)(90,149,168,287)(91,150,169,288)(92,151,170,289)(93,152,171,290)(94,153,172,291)(95,154,173,292)(96,155,174,293)(97,156,175,294)(98,157,176,295)(99,158,177,296)(100,159,178,297)(101,160,179,298)(102,121,180,299)(103,122,181,300)(104,123,182,301)(105,124,183,302)(106,125,184,303)(107,126,185,304)(108,127,186,305)(109,128,187,306)(110,129,188,307)(111,130,189,308)(112,131,190,309)(113,132,191,310)(114,133,192,311)(115,134,193,312)(116,135,194,313)(117,136,195,314)(118,137,196,315)(119,138,197,316)(120,139,198,317), (1,115,201,193)(2,94,202,172)(3,113,203,191)(4,92,204,170)(5,111,205,189)(6,90,206,168)(7,109,207,187)(8,88,208,166)(9,107,209,185)(10,86,210,164)(11,105,211,183)(12,84,212,162)(13,103,213,181)(14,82,214,200)(15,101,215,179)(16,120,216,198)(17,99,217,177)(18,118,218,196)(19,97,219,175)(20,116,220,194)(21,95,221,173)(22,114,222,192)(23,93,223,171)(24,112,224,190)(25,91,225,169)(26,110,226,188)(27,89,227,167)(28,108,228,186)(29,87,229,165)(30,106,230,184)(31,85,231,163)(32,104,232,182)(33,83,233,161)(34,102,234,180)(35,81,235,199)(36,100,236,178)(37,119,237,197)(38,98,238,176)(39,117,239,195)(40,96,240,174)(41,306,252,128)(42,285,253,147)(43,304,254,126)(44,283,255,145)(45,302,256,124)(46,281,257,143)(47,300,258,122)(48,319,259,141)(49,298,260,160)(50,317,261,139)(51,296,262,158)(52,315,263,137)(53,294,264,156)(54,313,265,135)(55,292,266,154)(56,311,267,133)(57,290,268,152)(58,309,269,131)(59,288,270,150)(60,307,271,129)(61,286,272,148)(62,305,273,127)(63,284,274,146)(64,303,275,125)(65,282,276,144)(66,301,277,123)(67,320,278,142)(68,299,279,121)(69,318,280,140)(70,297,241,159)(71,316,242,138)(72,295,243,157)(73,314,244,136)(74,293,245,155)(75,312,246,134)(76,291,247,153)(77,310,248,132)(78,289,249,151)(79,308,250,130)(80,287,251,149) );
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)], [(1,75,201,246),(2,76,202,247),(3,77,203,248),(4,78,204,249),(5,79,205,250),(6,80,206,251),(7,41,207,252),(8,42,208,253),(9,43,209,254),(10,44,210,255),(11,45,211,256),(12,46,212,257),(13,47,213,258),(14,48,214,259),(15,49,215,260),(16,50,216,261),(17,51,217,262),(18,52,218,263),(19,53,219,264),(20,54,220,265),(21,55,221,266),(22,56,222,267),(23,57,223,268),(24,58,224,269),(25,59,225,270),(26,60,226,271),(27,61,227,272),(28,62,228,273),(29,63,229,274),(30,64,230,275),(31,65,231,276),(32,66,232,277),(33,67,233,278),(34,68,234,279),(35,69,235,280),(36,70,236,241),(37,71,237,242),(38,72,238,243),(39,73,239,244),(40,74,240,245),(81,140,199,318),(82,141,200,319),(83,142,161,320),(84,143,162,281),(85,144,163,282),(86,145,164,283),(87,146,165,284),(88,147,166,285),(89,148,167,286),(90,149,168,287),(91,150,169,288),(92,151,170,289),(93,152,171,290),(94,153,172,291),(95,154,173,292),(96,155,174,293),(97,156,175,294),(98,157,176,295),(99,158,177,296),(100,159,178,297),(101,160,179,298),(102,121,180,299),(103,122,181,300),(104,123,182,301),(105,124,183,302),(106,125,184,303),(107,126,185,304),(108,127,186,305),(109,128,187,306),(110,129,188,307),(111,130,189,308),(112,131,190,309),(113,132,191,310),(114,133,192,311),(115,134,193,312),(116,135,194,313),(117,136,195,314),(118,137,196,315),(119,138,197,316),(120,139,198,317)], [(1,115,201,193),(2,94,202,172),(3,113,203,191),(4,92,204,170),(5,111,205,189),(6,90,206,168),(7,109,207,187),(8,88,208,166),(9,107,209,185),(10,86,210,164),(11,105,211,183),(12,84,212,162),(13,103,213,181),(14,82,214,200),(15,101,215,179),(16,120,216,198),(17,99,217,177),(18,118,218,196),(19,97,219,175),(20,116,220,194),(21,95,221,173),(22,114,222,192),(23,93,223,171),(24,112,224,190),(25,91,225,169),(26,110,226,188),(27,89,227,167),(28,108,228,186),(29,87,229,165),(30,106,230,184),(31,85,231,163),(32,104,232,182),(33,83,233,161),(34,102,234,180),(35,81,235,199),(36,100,236,178),(37,119,237,197),(38,98,238,176),(39,117,239,195),(40,96,240,174),(41,306,252,128),(42,285,253,147),(43,304,254,126),(44,283,255,145),(45,302,256,124),(46,281,257,143),(47,300,258,122),(48,319,259,141),(49,298,260,160),(50,317,261,139),(51,296,262,158),(52,315,263,137),(53,294,264,156),(54,313,265,135),(55,292,266,154),(56,311,267,133),(57,290,268,152),(58,309,269,131),(59,288,270,150),(60,307,271,129),(61,286,272,148),(62,305,273,127),(63,284,274,146),(64,303,275,125),(65,282,276,144),(66,301,277,123),(67,320,278,142),(68,299,279,121),(69,318,280,140),(70,297,241,159),(71,316,242,138),(72,295,243,157),(73,314,244,136),(74,293,245,155),(75,312,246,134),(76,291,247,153),(77,310,248,132),(78,289,249,151),(79,308,250,130),(80,287,251,149)]])
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 |
| type | + | + | + | + | - | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | Q8 | D4 | D5 | SD16 | D10 | D10 | Dic10 | D20 | C40⋊C2 |
| kernel | C40⋊9Q8 | C40⋊6C4 | C4×C40 | C20⋊2Q8 | C40 | C2×C20 | C4×C8 | C20 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 4 | 2 | 2 | 8 | 2 | 4 | 16 | 8 | 32 |
Matrix representation of C40⋊9Q8 ►in GL4(𝔽41) generated by
| 26 | 15 | 0 | 0 |
| 26 | 26 | 0 | 0 |
| 0 | 0 | 16 | 14 |
| 0 | 0 | 29 | 28 |
| 0 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 39 | 32 |
| 0 | 0 | 37 | 2 |
| 38 | 21 | 0 | 0 |
| 21 | 3 | 0 | 0 |
| 0 | 0 | 21 | 3 |
| 0 | 0 | 3 | 20 |
G:=sub<GL(4,GF(41))| [26,26,0,0,15,26,0,0,0,0,16,29,0,0,14,28],[0,1,0,0,40,0,0,0,0,0,39,37,0,0,32,2],[38,21,0,0,21,3,0,0,0,0,21,3,0,0,3,20] >;
C40⋊9Q8 in GAP, Magma, Sage, TeX
C_{40}\rtimes_9Q_8 % in TeX
G:=Group("C40:9Q8"); // GroupNames label
G:=SmallGroup(320,307);
// by ID
G=gap.SmallGroup(320,307);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,120,254,58,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^19,c*b*c^-1=b^-1>;
// generators/relations