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## G = C40.15D4order 320 = 26·5

### 15th non-split extension by C40 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C40.15D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C40 — C40⋊5C4 — C40.15D4
 Lower central C5 — C10 — C20 — C40 — C40.15D4
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2×Q16

Generators and relations for C40.15D4
G = < a,b,c | a40=b4=1, c2=a35, bab-1=a-1, cac-1=a9, cbc-1=a15b-1 >

Subgroups: 190 in 58 conjugacy classes, 31 normal (27 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C16, C4⋊C4, C2×C8, Q16, Q16, C2×Q8, Dic5, C20, C20, C2×C10, C2.D8, C2×C16, C2×Q16, C40, C2×Dic5, C2×C20, C2×C20, C5×Q8, C2.Q32, C52C16, C4⋊Dic5, C2×C40, C5×Q16, C5×Q16, Q8×C10, C2×C52C16, C405C4, C10×Q16, C40.15D4
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D8, SD16, Dic5, D10, D4⋊C4, SD32, Q32, C2×Dic5, C5⋊D4, C2.Q32, D4⋊D5, D4.D5, C23.D5, C5⋊SD32, C5⋊Q32, D4⋊Dic5, C40.15D4

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

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - - - + + - image C1 C2 C2 C2 C4 D4 D4 D5 SD16 D8 D10 Dic5 SD32 Q32 C5⋊D4 C5⋊D4 D4.D5 D4⋊D5 C5⋊SD32 C5⋊Q32 kernel C40.15D4 C2×C5⋊2C16 C40⋊5C4 C10×Q16 C5×Q16 C40 C2×C20 C2×Q16 C20 C2×C10 C2×C8 Q16 C10 C10 C8 C2×C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 2 2 2 2 4 4 4 4 4 2 2 4 4

Matrix representation of C40.15D4 in GL5(𝔽241)

 240 0 0 0 0 0 1 52 0 0 0 189 189 0 0 0 0 0 0 148 0 0 0 184 219
,
 177 0 0 0 0 0 206 157 0 0 0 49 35 0 0 0 0 0 61 145 0 0 0 94 180
,
 177 0 0 0 0 0 5 132 0 0 0 113 236 0 0 0 0 0 112 92 0 0 0 173 183

`G:=sub<GL(5,GF(241))| [240,0,0,0,0,0,1,189,0,0,0,52,189,0,0,0,0,0,0,184,0,0,0,148,219],[177,0,0,0,0,0,206,49,0,0,0,157,35,0,0,0,0,0,61,94,0,0,0,145,180],[177,0,0,0,0,0,5,113,0,0,0,132,236,0,0,0,0,0,112,173,0,0,0,92,183] >;`

C40.15D4 in GAP, Magma, Sage, TeX

`C_{40}._{15}D_4`
`% in TeX`

`G:=Group("C40.15D4");`
`// GroupNames label`

`G:=SmallGroup(320,122);`
`// by ID`

`G=gap.SmallGroup(320,122);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,675,346,192,1684,851,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^40=b^4=1,c^2=a^35,b*a*b^-1=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^15*b^-1>;`
`// generators/relations`

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