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## G = Dic5.15C42order 320 = 26·5

### 1st non-split extension by Dic5 of C42 acting through Inn(Dic5)

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic5.15C42
G = < a,b,c,d | a10=c4=d4=1, b2=a5, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a5c >

Subgroups: 478 in 178 conjugacy classes, 91 normal (12 characteristic)
C1, C2, C2 [×6], C4 [×16], C22, C22 [×6], C5, C2×C4 [×6], C2×C4 [×24], C23, C10, C10 [×6], C42 [×12], C22×C4 [×3], C22×C4 [×4], Dic5 [×4], Dic5 [×6], C20 [×6], C2×C10, C2×C10 [×6], C2.C42, C2.C42 [×3], C2×C42 [×3], C2×Dic5 [×12], C2×Dic5 [×6], C2×C20 [×6], C2×C20 [×6], C22×C10, C424C4, C4×Dic5 [×12], C22×Dic5, C22×Dic5 [×3], C22×C20 [×3], C10.10C42 [×3], C5×C2.C42, C2×C4×Dic5 [×3], Dic5.15C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, D5, C42 [×4], C22×C4 [×3], C4○D4 [×4], D10 [×3], C2×C42, C42⋊C2 [×6], C4×D5 [×6], C22×D5, C424C4, C2×C4×D5 [×3], D42D5 [×3], Q82D5, D5×C42, C23.11D10 [×3], C4⋊C47D5 [×3], Dic5.15C42

Smallest permutation representation of Dic5.15C42
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)
(1 166 6 161)(2 165 7 170)(3 164 8 169)(4 163 9 168)(5 162 10 167)(11 174 16 179)(12 173 17 178)(13 172 18 177)(14 171 19 176)(15 180 20 175)(21 184 26 189)(22 183 27 188)(23 182 28 187)(24 181 29 186)(25 190 30 185)(31 194 36 199)(32 193 37 198)(33 192 38 197)(34 191 39 196)(35 200 40 195)(41 204 46 209)(42 203 47 208)(43 202 48 207)(44 201 49 206)(45 210 50 205)(51 214 56 219)(52 213 57 218)(53 212 58 217)(54 211 59 216)(55 220 60 215)(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)(81 244 86 249)(82 243 87 248)(83 242 88 247)(84 241 89 246)(85 250 90 245)(91 254 96 259)(92 253 97 258)(93 252 98 257)(94 251 99 256)(95 260 100 255)(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)(121 284 126 289)(122 283 127 288)(123 282 128 287)(124 281 129 286)(125 290 130 285)(131 294 136 299)(132 293 137 298)(133 292 138 297)(134 291 139 296)(135 300 140 295)(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)
(1 99 19 89)(2 100 20 90)(3 91 11 81)(4 92 12 82)(5 93 13 83)(6 94 14 84)(7 95 15 85)(8 96 16 86)(9 97 17 87)(10 98 18 88)(21 111 31 101)(22 112 32 102)(23 113 33 103)(24 114 34 104)(25 115 35 105)(26 116 36 106)(27 117 37 107)(28 118 38 108)(29 119 39 109)(30 120 40 110)(41 131 51 121)(42 132 52 122)(43 133 53 123)(44 134 54 124)(45 135 55 125)(46 136 56 126)(47 137 57 127)(48 138 58 128)(49 139 59 129)(50 140 60 130)(61 151 71 141)(62 152 72 142)(63 153 73 143)(64 154 74 144)(65 155 75 145)(66 156 76 146)(67 157 77 147)(68 158 78 148)(69 159 79 149)(70 160 80 150)(161 251 171 241)(162 252 172 242)(163 253 173 243)(164 254 174 244)(165 255 175 245)(166 256 176 246)(167 257 177 247)(168 258 178 248)(169 259 179 249)(170 260 180 250)(181 271 191 261)(182 272 192 262)(183 273 193 263)(184 274 194 264)(185 275 195 265)(186 276 196 266)(187 277 197 267)(188 278 198 268)(189 279 199 269)(190 280 200 270)(201 291 211 281)(202 292 212 282)(203 293 213 283)(204 294 214 284)(205 295 215 285)(206 296 216 286)(207 297 217 287)(208 298 218 288)(209 299 219 289)(210 300 220 290)(221 311 231 301)(222 312 232 302)(223 313 233 303)(224 314 234 304)(225 315 235 305)(226 316 236 306)(227 317 237 307)(228 318 238 308)(229 319 239 309)(230 320 240 310)
(1 69 29 49)(2 70 30 50)(3 61 21 41)(4 62 22 42)(5 63 23 43)(6 64 24 44)(7 65 25 45)(8 66 26 46)(9 67 27 47)(10 68 28 48)(11 71 31 51)(12 72 32 52)(13 73 33 53)(14 74 34 54)(15 75 35 55)(16 76 36 56)(17 77 37 57)(18 78 38 58)(19 79 39 59)(20 80 40 60)(81 146 101 126)(82 147 102 127)(83 148 103 128)(84 149 104 129)(85 150 105 130)(86 141 106 121)(87 142 107 122)(88 143 108 123)(89 144 109 124)(90 145 110 125)(91 156 111 136)(92 157 112 137)(93 158 113 138)(94 159 114 139)(95 160 115 140)(96 151 116 131)(97 152 117 132)(98 153 118 133)(99 154 119 134)(100 155 120 135)(161 221 181 201)(162 222 182 202)(163 223 183 203)(164 224 184 204)(165 225 185 205)(166 226 186 206)(167 227 187 207)(168 228 188 208)(169 229 189 209)(170 230 190 210)(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)(241 306 261 286)(242 307 262 287)(243 308 263 288)(244 309 264 289)(245 310 265 290)(246 301 266 281)(247 302 267 282)(248 303 268 283)(249 304 269 284)(250 305 270 285)(251 316 271 296)(252 317 272 297)(253 318 273 298)(254 319 274 299)(255 320 275 300)(256 311 276 291)(257 312 277 292)(258 313 278 293)(259 314 279 294)(260 315 280 295)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4L 4M ··· 4T 4U ··· 4AF 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 4 ··· 4 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 ··· 2 5 ··· 5 10 ··· 10 2 2 2 ··· 2 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + - + image C1 C2 C2 C2 C4 D5 C4○D4 D10 C4×D5 D4⋊2D5 Q8⋊2D5 kernel Dic5.15C42 C10.10C42 C5×C2.C42 C2×C4×Dic5 C4×Dic5 C2.C42 C2×C10 C22×C4 C2×C4 C22 C22 # reps 1 3 1 3 24 2 8 6 24 6 2

Matrix representation of Dic5.15C42 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 22 18 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 15 0 0 0 0 0 40 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 37 3 0 0 0 0 36 4
,
 40 0 0 0 0 0 0 32 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 32 39 0 0 0 0 0 9

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,22,0,0,0,0,0,18,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,15,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,37,36,0,0,0,0,3,4],[40,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,39,9] >;

Dic5.15C42 in GAP, Magma, Sage, TeX

{\rm Dic}_5._{15}C_4^2
% in TeX

G:=Group("Dic5.15C4^2");
// GroupNames label

G:=SmallGroup(320,275);
// by ID

G=gap.SmallGroup(320,275);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,387,58,12550]);
// Polycyclic

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

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