Copied to
clipboard

## G = C4⋊Dic5⋊15C4order 320 = 26·5

### 4th semidirect product of C4⋊Dic5 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C4⋊Dic5⋊15C4
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C2×C4⋊Dic5 — C4⋊Dic5⋊15C4
 Lower central C5 — C2×C10 — C4⋊Dic5⋊15C4
 Upper central C1 — C23 — C2.C42

Generators and relations for C4⋊Dic515C4
G = < a,b,c,d | a4=b10=d4=1, c2=b5, ab=ba, cac-1=a-1, dad-1=a-1b5, cbc-1=b-1, bd=db, cd=dc >

Subgroups: 478 in 154 conjugacy classes, 69 normal (51 characteristic)
C1, C2, C4, C22, C5, C2×C4, C2×C4, C23, C10, C42, C4⋊C4, C22×C4, C22×C4, Dic5, C20, C2×C10, C2.C42, C2.C42, C2×C42, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.63C23, C4×Dic5, C10.D4, C4⋊Dic5, C22×Dic5, C22×C20, C10.10C42, C5×C2.C42, C2×C4×Dic5, C2×C10.D4, C2×C4⋊Dic5, C4⋊Dic515C4
Quotients:

Smallest permutation representation of C4⋊Dic515C4
Regular action on 320 points
Generators in S320
(1 102 22 96)(2 103 23 97)(3 104 24 98)(4 105 25 99)(5 106 26 100)(6 107 27 91)(7 108 28 92)(8 109 29 93)(9 110 30 94)(10 101 21 95)(11 239 319 248)(12 240 320 249)(13 231 311 250)(14 232 312 241)(15 233 313 242)(16 234 314 243)(17 235 315 244)(18 236 316 245)(19 237 317 246)(20 238 318 247)(31 127 47 111)(32 128 48 112)(33 129 49 113)(34 130 50 114)(35 121 41 115)(36 122 42 116)(37 123 43 117)(38 124 44 118)(39 125 45 119)(40 126 46 120)(51 147 67 131)(52 148 68 132)(53 149 69 133)(54 150 70 134)(55 141 61 135)(56 142 62 136)(57 143 63 137)(58 144 64 138)(59 145 65 139)(60 146 66 140)(71 170 87 151)(72 161 88 152)(73 162 89 153)(74 163 90 154)(75 164 81 155)(76 165 82 156)(77 166 83 157)(78 167 84 158)(79 168 85 159)(80 169 86 160)(171 251 190 270)(172 252 181 261)(173 253 182 262)(174 254 183 263)(175 255 184 264)(176 256 185 265)(177 257 186 266)(178 258 187 267)(179 259 188 268)(180 260 189 269)(191 271 210 290)(192 272 201 281)(193 273 202 282)(194 274 203 283)(195 275 204 284)(196 276 205 285)(197 277 206 286)(198 278 207 287)(199 279 208 288)(200 280 209 289)(211 291 230 310)(212 292 221 301)(213 293 222 302)(214 294 223 303)(215 295 224 304)(216 296 225 305)(217 297 226 306)(218 298 227 307)(219 299 228 308)(220 300 229 309)
(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)
(1 179 6 174)(2 178 7 173)(3 177 8 172)(4 176 9 171)(5 175 10 180)(11 170 16 165)(12 169 17 164)(13 168 18 163)(14 167 19 162)(15 166 20 161)(21 189 26 184)(22 188 27 183)(23 187 28 182)(24 186 29 181)(25 185 30 190)(31 194 36 199)(32 193 37 198)(33 192 38 197)(34 191 39 196)(35 200 40 195)(41 209 46 204)(42 208 47 203)(43 207 48 202)(44 206 49 201)(45 205 50 210)(51 214 56 219)(52 213 57 218)(53 212 58 217)(54 211 59 216)(55 220 60 215)(61 229 66 224)(62 228 67 223)(63 227 68 222)(64 226 69 221)(65 225 70 230)(71 234 76 239)(72 233 77 238)(73 232 78 237)(74 231 79 236)(75 240 80 235)(81 249 86 244)(82 248 87 243)(83 247 88 242)(84 246 89 241)(85 245 90 250)(91 254 96 259)(92 253 97 258)(93 252 98 257)(94 251 99 256)(95 260 100 255)(101 269 106 264)(102 268 107 263)(103 267 108 262)(104 266 109 261)(105 265 110 270)(111 274 116 279)(112 273 117 278)(113 272 118 277)(114 271 119 276)(115 280 120 275)(121 289 126 284)(122 288 127 283)(123 287 128 282)(124 286 129 281)(125 285 130 290)(131 294 136 299)(132 293 137 298)(133 292 138 297)(134 291 139 296)(135 300 140 295)(141 309 146 304)(142 308 147 303)(143 307 148 302)(144 306 149 301)(145 305 150 310)(151 314 156 319)(152 313 157 318)(153 312 158 317)(154 311 159 316)(155 320 160 315)
(1 76 36 56)(2 77 37 57)(3 78 38 58)(4 79 39 59)(5 80 40 60)(6 71 31 51)(7 72 32 52)(8 73 33 53)(9 74 34 54)(10 75 35 55)(11 274 308 254)(12 275 309 255)(13 276 310 256)(14 277 301 257)(15 278 302 258)(16 279 303 259)(17 280 304 260)(18 271 305 251)(19 272 306 252)(20 273 307 253)(21 81 41 61)(22 82 42 62)(23 83 43 63)(24 84 44 64)(25 85 45 65)(26 86 46 66)(27 87 47 67)(28 88 48 68)(29 89 49 69)(30 90 50 70)(91 165 111 142)(92 166 112 143)(93 167 113 144)(94 168 114 145)(95 169 115 146)(96 170 116 147)(97 161 117 148)(98 162 118 149)(99 163 119 150)(100 164 120 141)(101 160 121 140)(102 151 122 131)(103 152 123 132)(104 153 124 133)(105 154 125 134)(106 155 126 135)(107 156 127 136)(108 157 128 137)(109 158 129 138)(110 159 130 139)(171 231 191 211)(172 232 192 212)(173 233 193 213)(174 234 194 214)(175 235 195 215)(176 236 196 216)(177 237 197 217)(178 238 198 218)(179 239 199 219)(180 240 200 220)(181 241 201 221)(182 242 202 222)(183 243 203 223)(184 244 204 224)(185 245 205 225)(186 246 206 226)(187 247 207 227)(188 248 208 228)(189 249 209 229)(190 250 210 230)(261 317 281 297)(262 318 282 298)(263 319 283 299)(264 320 284 300)(265 311 285 291)(266 312 286 292)(267 313 287 293)(268 314 288 294)(269 315 289 295)(270 316 290 296)

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 4Q 4R 4S 4T 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 4 4 4 4 4 4 4 4 4 ··· 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 2 2 2 4 4 4 4 10 ··· 10 20 20 20 20 2 2 2 ··· 2 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + - + + - + - + image C1 C2 C2 C2 C2 C2 C4 D4 Q8 D5 C4○D4 D10 Dic10 C4×D5 C4○D20 D4×D5 D4⋊2D5 Q8⋊2D5 kernel C4⋊Dic5⋊15C4 C10.10C42 C5×C2.C42 C2×C4×Dic5 C2×C10.D4 C2×C4⋊Dic5 C4⋊Dic5 C2×Dic5 C2×C20 C2.C42 C2×C10 C22×C4 C2×C4 C2×C4 C22 C22 C22 C22 # reps 1 3 1 1 1 1 8 2 2 2 8 6 8 8 8 2 4 2

Matrix representation of C4⋊Dic515C4 in GL6(𝔽41)

 27 34 0 0 0 0 34 14 0 0 0 0 0 0 8 2 0 0 0 0 29 33 0 0 0 0 0 0 30 9 0 0 0 0 32 11
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 40 7
,
 0 9 0 0 0 0 32 0 0 0 0 0 0 0 32 29 0 0 0 0 0 9 0 0 0 0 0 0 24 40 0 0 0 0 3 17
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(41))| [27,34,0,0,0,0,34,14,0,0,0,0,0,0,8,29,0,0,0,0,2,33,0,0,0,0,0,0,30,32,0,0,0,0,9,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,7],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,32,0,0,0,0,0,29,9,0,0,0,0,0,0,24,3,0,0,0,0,40,17],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C4⋊Dic515C4 in GAP, Magma, Sage, TeX

C_4\rtimes {\rm Dic}_5\rtimes_{15}C_4
% in TeX

G:=Group("C4:Dic5:15C4");
// GroupNames label

G:=SmallGroup(320,281);
// by ID

G=gap.SmallGroup(320,281);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,701,344,387,58,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^10=d^4=1,c^2=b^5,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^5,c*b*c^-1=b^-1,b*d=d*b,c*d=d*c>;
// generators/relations

׿
×
𝔽