Copied to
clipboard

## G = C2×C40⋊6C4order 320 = 26·5

### Direct product of C2 and C40⋊6C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×C40⋊6C4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C4⋊Dic5 — C2×C4⋊Dic5 — C2×C40⋊6C4
 Lower central C5 — C10 — C20 — C2×C40⋊6C4
 Upper central C1 — C23 — C22×C4 — C22×C8

Generators and relations for C2×C406C4
G = < a,b,c | a2=b40=c4=1, ab=ba, ac=ca, cbc-1=b19 >

Subgroups: 430 in 130 conjugacy classes, 87 normal (21 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C4⋊C4, C2×C8, C22×C4, C22×C4, Dic5, C20, C20, C2×C10, C2×C10, C4.Q8, C2×C4⋊C4, C22×C8, C40, C2×Dic5, C2×C20, C2×C20, C22×C10, C2×C4.Q8, C4⋊Dic5, C4⋊Dic5, C2×C40, C22×Dic5, C22×C20, C406C4, C2×C4⋊Dic5, C22×C40, C2×C406C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, Dic5, D10, C4.Q8, C2×C4⋊C4, C2×SD16, Dic10, D20, C2×Dic5, C22×D5, C2×C4.Q8, C40⋊C2, C4⋊Dic5, C2×Dic10, C2×D20, C22×Dic5, C406C4, C2×C40⋊C2, C2×C4⋊Dic5, C2×C406C4

Smallest permutation representation of C2×C406C4
Regular action on 320 points
Generators in S320
(1 315)(2 316)(3 317)(4 318)(5 319)(6 320)(7 281)(8 282)(9 283)(10 284)(11 285)(12 286)(13 287)(14 288)(15 289)(16 290)(17 291)(18 292)(19 293)(20 294)(21 295)(22 296)(23 297)(24 298)(25 299)(26 300)(27 301)(28 302)(29 303)(30 304)(31 305)(32 306)(33 307)(34 308)(35 309)(36 310)(37 311)(38 312)(39 313)(40 314)(41 96)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(57 112)(58 113)(59 114)(60 115)(61 116)(62 117)(63 118)(64 119)(65 120)(66 81)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)(73 88)(74 89)(75 90)(76 91)(77 92)(78 93)(79 94)(80 95)(121 223)(122 224)(123 225)(124 226)(125 227)(126 228)(127 229)(128 230)(129 231)(130 232)(131 233)(132 234)(133 235)(134 236)(135 237)(136 238)(137 239)(138 240)(139 201)(140 202)(141 203)(142 204)(143 205)(144 206)(145 207)(146 208)(147 209)(148 210)(149 211)(150 212)(151 213)(152 214)(153 215)(154 216)(155 217)(156 218)(157 219)(158 220)(159 221)(160 222)(161 245)(162 246)(163 247)(164 248)(165 249)(166 250)(167 251)(168 252)(169 253)(170 254)(171 255)(172 256)(173 257)(174 258)(175 259)(176 260)(177 261)(178 262)(179 263)(180 264)(181 265)(182 266)(183 267)(184 268)(185 269)(186 270)(187 271)(188 272)(189 273)(190 274)(191 275)(192 276)(193 277)(194 278)(195 279)(196 280)(197 241)(198 242)(199 243)(200 244)
(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 70 147 175)(2 49 148 194)(3 68 149 173)(4 47 150 192)(5 66 151 171)(6 45 152 190)(7 64 153 169)(8 43 154 188)(9 62 155 167)(10 41 156 186)(11 60 157 165)(12 79 158 184)(13 58 159 163)(14 77 160 182)(15 56 121 161)(16 75 122 180)(17 54 123 199)(18 73 124 178)(19 52 125 197)(20 71 126 176)(21 50 127 195)(22 69 128 174)(23 48 129 193)(24 67 130 172)(25 46 131 191)(26 65 132 170)(27 44 133 189)(28 63 134 168)(29 42 135 187)(30 61 136 166)(31 80 137 185)(32 59 138 164)(33 78 139 183)(34 57 140 162)(35 76 141 181)(36 55 142 200)(37 74 143 179)(38 53 144 198)(39 72 145 177)(40 51 146 196)(81 213 255 319)(82 232 256 298)(83 211 257 317)(84 230 258 296)(85 209 259 315)(86 228 260 294)(87 207 261 313)(88 226 262 292)(89 205 263 311)(90 224 264 290)(91 203 265 309)(92 222 266 288)(93 201 267 307)(94 220 268 286)(95 239 269 305)(96 218 270 284)(97 237 271 303)(98 216 272 282)(99 235 273 301)(100 214 274 320)(101 233 275 299)(102 212 276 318)(103 231 277 297)(104 210 278 316)(105 229 279 295)(106 208 280 314)(107 227 241 293)(108 206 242 312)(109 225 243 291)(110 204 244 310)(111 223 245 289)(112 202 246 308)(113 221 247 287)(114 240 248 306)(115 219 249 285)(116 238 250 304)(117 217 251 283)(118 236 252 302)(119 215 253 281)(120 234 254 300)

G:=sub<Sym(320)| (1,315)(2,316)(3,317)(4,318)(5,319)(6,320)(7,281)(8,282)(9,283)(10,284)(11,285)(12,286)(13,287)(14,288)(15,289)(16,290)(17,291)(18,292)(19,293)(20,294)(21,295)(22,296)(23,297)(24,298)(25,299)(26,300)(27,301)(28,302)(29,303)(30,304)(31,305)(32,306)(33,307)(34,308)(35,309)(36,310)(37,311)(38,312)(39,313)(40,314)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(57,112)(58,113)(59,114)(60,115)(61,116)(62,117)(63,118)(64,119)(65,120)(66,81)(67,82)(68,83)(69,84)(70,85)(71,86)(72,87)(73,88)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95)(121,223)(122,224)(123,225)(124,226)(125,227)(126,228)(127,229)(128,230)(129,231)(130,232)(131,233)(132,234)(133,235)(134,236)(135,237)(136,238)(137,239)(138,240)(139,201)(140,202)(141,203)(142,204)(143,205)(144,206)(145,207)(146,208)(147,209)(148,210)(149,211)(150,212)(151,213)(152,214)(153,215)(154,216)(155,217)(156,218)(157,219)(158,220)(159,221)(160,222)(161,245)(162,246)(163,247)(164,248)(165,249)(166,250)(167,251)(168,252)(169,253)(170,254)(171,255)(172,256)(173,257)(174,258)(175,259)(176,260)(177,261)(178,262)(179,263)(180,264)(181,265)(182,266)(183,267)(184,268)(185,269)(186,270)(187,271)(188,272)(189,273)(190,274)(191,275)(192,276)(193,277)(194,278)(195,279)(196,280)(197,241)(198,242)(199,243)(200,244), (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), (1,70,147,175)(2,49,148,194)(3,68,149,173)(4,47,150,192)(5,66,151,171)(6,45,152,190)(7,64,153,169)(8,43,154,188)(9,62,155,167)(10,41,156,186)(11,60,157,165)(12,79,158,184)(13,58,159,163)(14,77,160,182)(15,56,121,161)(16,75,122,180)(17,54,123,199)(18,73,124,178)(19,52,125,197)(20,71,126,176)(21,50,127,195)(22,69,128,174)(23,48,129,193)(24,67,130,172)(25,46,131,191)(26,65,132,170)(27,44,133,189)(28,63,134,168)(29,42,135,187)(30,61,136,166)(31,80,137,185)(32,59,138,164)(33,78,139,183)(34,57,140,162)(35,76,141,181)(36,55,142,200)(37,74,143,179)(38,53,144,198)(39,72,145,177)(40,51,146,196)(81,213,255,319)(82,232,256,298)(83,211,257,317)(84,230,258,296)(85,209,259,315)(86,228,260,294)(87,207,261,313)(88,226,262,292)(89,205,263,311)(90,224,264,290)(91,203,265,309)(92,222,266,288)(93,201,267,307)(94,220,268,286)(95,239,269,305)(96,218,270,284)(97,237,271,303)(98,216,272,282)(99,235,273,301)(100,214,274,320)(101,233,275,299)(102,212,276,318)(103,231,277,297)(104,210,278,316)(105,229,279,295)(106,208,280,314)(107,227,241,293)(108,206,242,312)(109,225,243,291)(110,204,244,310)(111,223,245,289)(112,202,246,308)(113,221,247,287)(114,240,248,306)(115,219,249,285)(116,238,250,304)(117,217,251,283)(118,236,252,302)(119,215,253,281)(120,234,254,300)>;

G:=Group( (1,315)(2,316)(3,317)(4,318)(5,319)(6,320)(7,281)(8,282)(9,283)(10,284)(11,285)(12,286)(13,287)(14,288)(15,289)(16,290)(17,291)(18,292)(19,293)(20,294)(21,295)(22,296)(23,297)(24,298)(25,299)(26,300)(27,301)(28,302)(29,303)(30,304)(31,305)(32,306)(33,307)(34,308)(35,309)(36,310)(37,311)(38,312)(39,313)(40,314)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(57,112)(58,113)(59,114)(60,115)(61,116)(62,117)(63,118)(64,119)(65,120)(66,81)(67,82)(68,83)(69,84)(70,85)(71,86)(72,87)(73,88)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95)(121,223)(122,224)(123,225)(124,226)(125,227)(126,228)(127,229)(128,230)(129,231)(130,232)(131,233)(132,234)(133,235)(134,236)(135,237)(136,238)(137,239)(138,240)(139,201)(140,202)(141,203)(142,204)(143,205)(144,206)(145,207)(146,208)(147,209)(148,210)(149,211)(150,212)(151,213)(152,214)(153,215)(154,216)(155,217)(156,218)(157,219)(158,220)(159,221)(160,222)(161,245)(162,246)(163,247)(164,248)(165,249)(166,250)(167,251)(168,252)(169,253)(170,254)(171,255)(172,256)(173,257)(174,258)(175,259)(176,260)(177,261)(178,262)(179,263)(180,264)(181,265)(182,266)(183,267)(184,268)(185,269)(186,270)(187,271)(188,272)(189,273)(190,274)(191,275)(192,276)(193,277)(194,278)(195,279)(196,280)(197,241)(198,242)(199,243)(200,244), (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), (1,70,147,175)(2,49,148,194)(3,68,149,173)(4,47,150,192)(5,66,151,171)(6,45,152,190)(7,64,153,169)(8,43,154,188)(9,62,155,167)(10,41,156,186)(11,60,157,165)(12,79,158,184)(13,58,159,163)(14,77,160,182)(15,56,121,161)(16,75,122,180)(17,54,123,199)(18,73,124,178)(19,52,125,197)(20,71,126,176)(21,50,127,195)(22,69,128,174)(23,48,129,193)(24,67,130,172)(25,46,131,191)(26,65,132,170)(27,44,133,189)(28,63,134,168)(29,42,135,187)(30,61,136,166)(31,80,137,185)(32,59,138,164)(33,78,139,183)(34,57,140,162)(35,76,141,181)(36,55,142,200)(37,74,143,179)(38,53,144,198)(39,72,145,177)(40,51,146,196)(81,213,255,319)(82,232,256,298)(83,211,257,317)(84,230,258,296)(85,209,259,315)(86,228,260,294)(87,207,261,313)(88,226,262,292)(89,205,263,311)(90,224,264,290)(91,203,265,309)(92,222,266,288)(93,201,267,307)(94,220,268,286)(95,239,269,305)(96,218,270,284)(97,237,271,303)(98,216,272,282)(99,235,273,301)(100,214,274,320)(101,233,275,299)(102,212,276,318)(103,231,277,297)(104,210,278,316)(105,229,279,295)(106,208,280,314)(107,227,241,293)(108,206,242,312)(109,225,243,291)(110,204,244,310)(111,223,245,289)(112,202,246,308)(113,221,247,287)(114,240,248,306)(115,219,249,285)(116,238,250,304)(117,217,251,283)(118,236,252,302)(119,215,253,281)(120,234,254,300) );

G=PermutationGroup([[(1,315),(2,316),(3,317),(4,318),(5,319),(6,320),(7,281),(8,282),(9,283),(10,284),(11,285),(12,286),(13,287),(14,288),(15,289),(16,290),(17,291),(18,292),(19,293),(20,294),(21,295),(22,296),(23,297),(24,298),(25,299),(26,300),(27,301),(28,302),(29,303),(30,304),(31,305),(32,306),(33,307),(34,308),(35,309),(36,310),(37,311),(38,312),(39,313),(40,314),(41,96),(42,97),(43,98),(44,99),(45,100),(46,101),(47,102),(48,103),(49,104),(50,105),(51,106),(52,107),(53,108),(54,109),(55,110),(56,111),(57,112),(58,113),(59,114),(60,115),(61,116),(62,117),(63,118),(64,119),(65,120),(66,81),(67,82),(68,83),(69,84),(70,85),(71,86),(72,87),(73,88),(74,89),(75,90),(76,91),(77,92),(78,93),(79,94),(80,95),(121,223),(122,224),(123,225),(124,226),(125,227),(126,228),(127,229),(128,230),(129,231),(130,232),(131,233),(132,234),(133,235),(134,236),(135,237),(136,238),(137,239),(138,240),(139,201),(140,202),(141,203),(142,204),(143,205),(144,206),(145,207),(146,208),(147,209),(148,210),(149,211),(150,212),(151,213),(152,214),(153,215),(154,216),(155,217),(156,218),(157,219),(158,220),(159,221),(160,222),(161,245),(162,246),(163,247),(164,248),(165,249),(166,250),(167,251),(168,252),(169,253),(170,254),(171,255),(172,256),(173,257),(174,258),(175,259),(176,260),(177,261),(178,262),(179,263),(180,264),(181,265),(182,266),(183,267),(184,268),(185,269),(186,270),(187,271),(188,272),(189,273),(190,274),(191,275),(192,276),(193,277),(194,278),(195,279),(196,280),(197,241),(198,242),(199,243),(200,244)], [(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)], [(1,70,147,175),(2,49,148,194),(3,68,149,173),(4,47,150,192),(5,66,151,171),(6,45,152,190),(7,64,153,169),(8,43,154,188),(9,62,155,167),(10,41,156,186),(11,60,157,165),(12,79,158,184),(13,58,159,163),(14,77,160,182),(15,56,121,161),(16,75,122,180),(17,54,123,199),(18,73,124,178),(19,52,125,197),(20,71,126,176),(21,50,127,195),(22,69,128,174),(23,48,129,193),(24,67,130,172),(25,46,131,191),(26,65,132,170),(27,44,133,189),(28,63,134,168),(29,42,135,187),(30,61,136,166),(31,80,137,185),(32,59,138,164),(33,78,139,183),(34,57,140,162),(35,76,141,181),(36,55,142,200),(37,74,143,179),(38,53,144,198),(39,72,145,177),(40,51,146,196),(81,213,255,319),(82,232,256,298),(83,211,257,317),(84,230,258,296),(85,209,259,315),(86,228,260,294),(87,207,261,313),(88,226,262,292),(89,205,263,311),(90,224,264,290),(91,203,265,309),(92,222,266,288),(93,201,267,307),(94,220,268,286),(95,239,269,305),(96,218,270,284),(97,237,271,303),(98,216,272,282),(99,235,273,301),(100,214,274,320),(101,233,275,299),(102,212,276,318),(103,231,277,297),(104,210,278,316),(105,229,279,295),(106,208,280,314),(107,227,241,293),(108,206,242,312),(109,225,243,291),(110,204,244,310),(111,223,245,289),(112,202,246,308),(113,221,247,287),(114,240,248,306),(115,219,249,285),(116,238,250,304),(117,217,251,283),(118,236,252,302),(119,215,253,281),(120,234,254,300)]])

92 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E ··· 4L 5A 5B 8A ··· 8H 10A ··· 10N 20A ··· 20P 40A ··· 40AF order 1 2 ··· 2 4 4 4 4 4 ··· 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 ··· 1 2 2 2 2 20 ··· 20 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

92 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + + - + + - + + image C1 C2 C2 C2 C4 D4 Q8 D4 D5 SD16 Dic5 D10 D10 Dic10 D20 D20 C40⋊C2 kernel C2×C40⋊6C4 C40⋊6C4 C2×C4⋊Dic5 C22×C40 C2×C40 C2×C20 C2×C20 C22×C10 C22×C8 C2×C10 C2×C8 C2×C8 C22×C4 C2×C4 C2×C4 C23 C22 # reps 1 4 2 1 8 1 2 1 2 8 8 4 2 8 4 4 32

Matrix representation of C2×C406C4 in GL5(𝔽41)

 1 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 27 14 0 0 0 27 11 0 0 0 0 0 18 28 0 0 0 13 27
,
 32 0 0 0 0 0 32 30 0 0 0 11 9 0 0 0 0 0 11 27 0 0 0 32 30

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,27,27,0,0,0,14,11,0,0,0,0,0,18,13,0,0,0,28,27],[32,0,0,0,0,0,32,11,0,0,0,30,9,0,0,0,0,0,11,32,0,0,0,27,30] >;

C2×C406C4 in GAP, Magma, Sage, TeX

C_2\times C_{40}\rtimes_6C_4
% in TeX

G:=Group("C2xC40:6C4");
// GroupNames label

G:=SmallGroup(320,731);
// by ID

G=gap.SmallGroup(320,731);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,100,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^2=b^40=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^19>;
// generators/relations

׿
×
𝔽