Copied to
clipboard

## G = C40.78D4order 320 = 26·5

### 1st non-split extension by C40 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C40.78D4
 Chief series C1 — C5 — C10 — C20 — C40 — C2×C40 — C40⋊5C4 — C40.78D4
 Lower central C5 — C10 — C20 — C40 — C40.78D4
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2×C16

Generators and relations for C40.78D4
G = < a,b,c | a40=b4=1, c2=a20, bab-1=cac-1=a-1, cbc-1=a25b-1 >

Subgroups: 286 in 58 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C16, C4⋊C4, C2×C8, Q16, C2×Q8, Dic5, C20, C2×C10, C2.D8, C2×C16, C2×Q16, C40, Dic10, C2×Dic5, C2×C20, C2.Q32, C80, Dic20, Dic20, C4⋊Dic5, C2×C40, C2×Dic10, C405C4, C2×C80, C2×Dic20, C40.78D4
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D8, SD16, D10, D4⋊C4, SD32, Q32, C4×D5, D20, C5⋊D4, C2.Q32, C40⋊C2, D40, D10⋊C4, C16⋊D5, Dic40, D205C4, C40.78D4

Smallest permutation representation of C40.78D4
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)
(1 183 83 264)(2 182 84 263)(3 181 85 262)(4 180 86 261)(5 179 87 260)(6 178 88 259)(7 177 89 258)(8 176 90 257)(9 175 91 256)(10 174 92 255)(11 173 93 254)(12 172 94 253)(13 171 95 252)(14 170 96 251)(15 169 97 250)(16 168 98 249)(17 167 99 248)(18 166 100 247)(19 165 101 246)(20 164 102 245)(21 163 103 244)(22 162 104 243)(23 161 105 242)(24 200 106 241)(25 199 107 280)(26 198 108 279)(27 197 109 278)(28 196 110 277)(29 195 111 276)(30 194 112 275)(31 193 113 274)(32 192 114 273)(33 191 115 272)(34 190 116 271)(35 189 117 270)(36 188 118 269)(37 187 119 268)(38 186 120 267)(39 185 81 266)(40 184 82 265)(41 313 144 231)(42 312 145 230)(43 311 146 229)(44 310 147 228)(45 309 148 227)(46 308 149 226)(47 307 150 225)(48 306 151 224)(49 305 152 223)(50 304 153 222)(51 303 154 221)(52 302 155 220)(53 301 156 219)(54 300 157 218)(55 299 158 217)(56 298 159 216)(57 297 160 215)(58 296 121 214)(59 295 122 213)(60 294 123 212)(61 293 124 211)(62 292 125 210)(63 291 126 209)(64 290 127 208)(65 289 128 207)(66 288 129 206)(67 287 130 205)(68 286 131 204)(69 285 132 203)(70 284 133 202)(71 283 134 201)(72 282 135 240)(73 281 136 239)(74 320 137 238)(75 319 138 237)(76 318 139 236)(77 317 140 235)(78 316 141 234)(79 315 142 233)(80 314 143 232)
(1 70 21 50)(2 69 22 49)(3 68 23 48)(4 67 24 47)(5 66 25 46)(6 65 26 45)(7 64 27 44)(8 63 28 43)(9 62 29 42)(10 61 30 41)(11 60 31 80)(12 59 32 79)(13 58 33 78)(14 57 34 77)(15 56 35 76)(16 55 36 75)(17 54 37 74)(18 53 38 73)(19 52 39 72)(20 51 40 71)(81 135 101 155)(82 134 102 154)(83 133 103 153)(84 132 104 152)(85 131 105 151)(86 130 106 150)(87 129 107 149)(88 128 108 148)(89 127 109 147)(90 126 110 146)(91 125 111 145)(92 124 112 144)(93 123 113 143)(94 122 114 142)(95 121 115 141)(96 160 116 140)(97 159 117 139)(98 158 118 138)(99 157 119 137)(100 156 120 136)(161 239 181 219)(162 238 182 218)(163 237 183 217)(164 236 184 216)(165 235 185 215)(166 234 186 214)(167 233 187 213)(168 232 188 212)(169 231 189 211)(170 230 190 210)(171 229 191 209)(172 228 192 208)(173 227 193 207)(174 226 194 206)(175 225 195 205)(176 224 196 204)(177 223 197 203)(178 222 198 202)(179 221 199 201)(180 220 200 240)(241 282 261 302)(242 281 262 301)(243 320 263 300)(244 319 264 299)(245 318 265 298)(246 317 266 297)(247 316 267 296)(248 315 268 295)(249 314 269 294)(250 313 270 293)(251 312 271 292)(252 311 272 291)(253 310 273 290)(254 309 274 289)(255 308 275 288)(256 307 276 287)(257 306 277 286)(258 305 278 285)(259 304 279 284)(260 303 280 283)```

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

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

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

86 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 8C 8D 10A ··· 10F 16A ··· 16H 20A ··· 20H 40A ··· 40P 80A ··· 80AF order 1 2 2 2 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 16 ··· 16 20 ··· 20 40 ··· 40 80 ··· 80 size 1 1 1 1 2 2 40 40 40 40 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

86 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + - + + - image C1 C2 C2 C2 C4 D4 D4 D5 SD16 D8 D10 SD32 Q32 C4×D5 C5⋊D4 D20 C40⋊C2 D40 C16⋊D5 Dic40 kernel C40.78D4 C40⋊5C4 C2×C80 C2×Dic20 Dic20 C40 C2×C20 C2×C16 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 8 8 16 16

Matrix representation of C40.78D4 in GL4(𝔽241) generated by

 240 51 0 0 190 190 0 0 0 0 9 56 0 0 185 214
,
 132 57 0 0 41 109 0 0 0 0 161 131 0 0 148 80
,
 232 33 0 0 56 9 0 0 0 0 214 106 0 0 175 27
`G:=sub<GL(4,GF(241))| [240,190,0,0,51,190,0,0,0,0,9,185,0,0,56,214],[132,41,0,0,57,109,0,0,0,0,161,148,0,0,131,80],[232,56,0,0,33,9,0,0,0,0,214,175,0,0,106,27] >;`

C40.78D4 in GAP, Magma, Sage, TeX

`C_{40}._{78}D_4`
`% in TeX`

`G:=Group("C40.78D4");`
`// GroupNames label`

`G:=SmallGroup(320,61);`
`// by ID`

`G=gap.SmallGroup(320,61);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,280,85,204,422,268,1684,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^40=b^4=1,c^2=a^20,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^25*b^-1>;`
`// generators/relations`

׿
×
𝔽