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## G = C4⋊C4⋊5Dic5order 320 = 26·5

### 3rd semidirect product of C4⋊C4 and Dic5 acting via Dic5/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C4⋊C4⋊5Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C2×C4×Dic5 — C4⋊C4⋊5Dic5
 Lower central C5 — C2×C10 — C4⋊C4⋊5Dic5
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for C4⋊C45Dic5
G = < a,b,c,d | a4=b4=c10=1, d2=c5, bab-1=a-1, ac=ca, dad-1=ab2, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 462 in 154 conjugacy classes, 75 normal (51 characteristic)
C1, C2 [×7], C4 [×12], C22 [×7], C5, C2×C4 [×6], C2×C4 [×20], C23, C10 [×7], C42 [×2], C4⋊C4 [×4], C4⋊C4 [×2], C22×C4 [×3], C22×C4 [×4], Dic5 [×6], C20 [×6], C2×C10 [×7], C2.C42 [×4], C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×Dic5 [×4], C2×Dic5 [×10], C2×C20 [×6], C2×C20 [×6], C22×C10, C23.63C23, C4×Dic5 [×2], C4⋊Dic5 [×2], C5×C4⋊C4 [×4], C22×Dic5 [×4], C22×C20 [×3], C10.10C42 [×4], C2×C4×Dic5, C2×C4⋊Dic5, C10×C4⋊C4, C4⋊C45Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C22×C4, C2×D4, C2×Q8, C4○D4 [×4], Dic5 [×4], D10 [×3], C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, C2×Dic5 [×6], C22×D5, C23.63C23, C4○D20 [×2], D4×D5, D42D5, Q8×D5, Q82D5, C22×Dic5, Dic5.Q8, D10.13D4, D10⋊Q8, C4⋊C4⋊D5, C23.21D10, D4×Dic5, Q8×Dic5, C4⋊C45Dic5

Smallest permutation representation of C4⋊C45Dic5
Regular action on 320 points
Generators in S320
(1 83 39 61)(2 84 40 62)(3 85 31 63)(4 86 32 64)(5 87 33 65)(6 88 34 66)(7 89 35 67)(8 90 36 68)(9 81 37 69)(10 82 38 70)(11 71 45 52)(12 72 46 53)(13 73 47 54)(14 74 48 55)(15 75 49 56)(16 76 50 57)(17 77 41 58)(18 78 42 59)(19 79 43 60)(20 80 44 51)(21 264 309 287)(22 265 310 288)(23 266 301 289)(24 267 302 290)(25 268 303 281)(26 269 304 282)(27 270 305 283)(28 261 306 284)(29 262 307 285)(30 263 308 286)(91 142 120 161)(92 143 111 162)(93 144 112 163)(94 145 113 164)(95 146 114 165)(96 147 115 166)(97 148 116 167)(98 149 117 168)(99 150 118 169)(100 141 119 170)(101 140 123 159)(102 131 124 160)(103 132 125 151)(104 133 126 152)(105 134 127 153)(106 135 128 154)(107 136 129 155)(108 137 130 156)(109 138 121 157)(110 139 122 158)(171 232 197 216)(172 233 198 217)(173 234 199 218)(174 235 200 219)(175 236 191 220)(176 237 192 211)(177 238 193 212)(178 239 194 213)(179 240 195 214)(180 231 196 215)(181 249 204 226)(182 250 205 227)(183 241 206 228)(184 242 207 229)(185 243 208 230)(186 244 209 221)(187 245 210 222)(188 246 201 223)(189 247 202 224)(190 248 203 225)(251 296 277 312)(252 297 278 313)(253 298 279 314)(254 299 280 315)(255 300 271 316)(256 291 272 317)(257 292 273 318)(258 293 274 319)(259 294 275 320)(260 295 276 311)
(1 101 14 95)(2 102 15 96)(3 103 16 97)(4 104 17 98)(5 105 18 99)(6 106 19 100)(7 107 20 91)(8 108 11 92)(9 109 12 93)(10 110 13 94)(21 247 316 231)(22 248 317 232)(23 249 318 233)(24 250 319 234)(25 241 320 235)(26 242 311 236)(27 243 312 237)(28 244 313 238)(29 245 314 239)(30 246 315 240)(31 125 50 116)(32 126 41 117)(33 127 42 118)(34 128 43 119)(35 129 44 120)(36 130 45 111)(37 121 46 112)(38 122 47 113)(39 123 48 114)(40 124 49 115)(51 142 67 136)(52 143 68 137)(53 144 69 138)(54 145 70 139)(55 146 61 140)(56 147 62 131)(57 148 63 132)(58 149 64 133)(59 150 65 134)(60 141 66 135)(71 162 90 156)(72 163 81 157)(73 164 82 158)(74 165 83 159)(75 166 84 160)(76 167 85 151)(77 168 86 152)(78 169 87 153)(79 170 88 154)(80 161 89 155)(171 265 190 256)(172 266 181 257)(173 267 182 258)(174 268 183 259)(175 269 184 260)(176 270 185 251)(177 261 186 252)(178 262 187 253)(179 263 188 254)(180 264 189 255)(191 282 207 276)(192 283 208 277)(193 284 209 278)(194 285 210 279)(195 286 201 280)(196 287 202 271)(197 288 203 272)(198 289 204 273)(199 290 205 274)(200 281 206 275)(211 305 230 296)(212 306 221 297)(213 307 222 298)(214 308 223 299)(215 309 224 300)(216 310 225 291)(217 301 226 292)(218 302 227 293)(219 303 228 294)(220 304 229 295)
(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)
(1 180 6 175)(2 179 7 174)(3 178 8 173)(4 177 9 172)(5 176 10 171)(11 182 16 187)(12 181 17 186)(13 190 18 185)(14 189 19 184)(15 188 20 183)(21 170 26 165)(22 169 27 164)(23 168 28 163)(24 167 29 162)(25 166 30 161)(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)(51 219 56 214)(52 218 57 213)(53 217 58 212)(54 216 59 211)(55 215 60 220)(61 224 66 229)(62 223 67 228)(63 222 68 227)(64 221 69 226)(65 230 70 225)(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)(91 259 96 254)(92 258 97 253)(93 257 98 252)(94 256 99 251)(95 255 100 260)(101 264 106 269)(102 263 107 268)(103 262 108 267)(104 261 109 266)(105 270 110 265)(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)(131 299 136 294)(132 298 137 293)(133 297 138 292)(134 296 139 291)(135 295 140 300)(141 304 146 309)(142 303 147 308)(143 302 148 307)(144 301 149 306)(145 310 150 305)(151 314 156 319)(152 313 157 318)(153 312 158 317)(154 311 159 316)(155 320 160 315)

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

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

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

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 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - + - + + - - + image C1 C2 C2 C2 C2 C4 D4 Q8 D5 C4○D4 Dic5 D10 C4○D20 D4×D5 D4⋊2D5 Q8×D5 Q8⋊2D5 kernel C4⋊C4⋊5Dic5 C10.10C42 C2×C4×Dic5 C2×C4⋊Dic5 C10×C4⋊C4 C5×C4⋊C4 C2×Dic5 C2×Dic5 C2×C4⋊C4 C2×C10 C4⋊C4 C22×C4 C22 C22 C22 C22 C22 # reps 1 4 1 1 1 8 2 2 2 8 8 6 16 2 2 2 2

Matrix representation of C4⋊C45Dic5 in GL6(𝔽41)

 24 35 0 0 0 0 7 17 0 0 0 0 0 0 18 1 0 0 0 0 5 23 0 0 0 0 0 0 35 5 0 0 0 0 9 6
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 0 23 0 0 0 0 16 0
,
 1 40 0 0 0 0 8 34 0 0 0 0 0 0 35 1 0 0 0 0 5 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 21 0 0 0 0 37 1 0 0 0 0 0 0 28 14 0 0 0 0 29 13 0 0 0 0 0 0 0 18 0 0 0 0 25 0

G:=sub<GL(6,GF(41))| [24,7,0,0,0,0,35,17,0,0,0,0,0,0,18,5,0,0,0,0,1,23,0,0,0,0,0,0,35,9,0,0,0,0,5,6],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,16,0,0,0,0,23,0],[1,8,0,0,0,0,40,34,0,0,0,0,0,0,35,5,0,0,0,0,1,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,37,0,0,0,0,21,1,0,0,0,0,0,0,28,29,0,0,0,0,14,13,0,0,0,0,0,0,0,25,0,0,0,0,18,0] >;

C4⋊C45Dic5 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\rtimes_5{\rm Dic}_5
% in TeX

G:=Group("C4:C4:5Dic5");
// GroupNames label

G:=SmallGroup(320,608);
// by ID

G=gap.SmallGroup(320,608);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,422,387,184,12550]);
// Polycyclic

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

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