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## G = Dic5⋊2C42order 320 = 26·5

### 1st semidirect product of Dic5 and C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — Dic5⋊2C42
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C2×C4×Dic5 — Dic5⋊2C42
 Lower central C5 — C10 — Dic5⋊2C42
 Upper central C1 — C23 — C2.C42

Generators and relations for Dic52C42
G = < a,b,c,d | a10=c4=d4=1, b2=a5, bab-1=cac-1=a-1, ad=da, bc=cb, dbd-1=a5b, cd=dc >

Subgroups: 526 in 194 conjugacy classes, 99 normal (25 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C42, C4⋊C4, C22×C4, C22×C4, C22×C4, Dic5, Dic5, C20, C2×C10, C2×C10, C2.C42, C2.C42, C2×C42, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C4×C4⋊C4, C4×Dic5, C4×Dic5, C10.D4, C22×Dic5, C22×Dic5, C22×C20, C22×C20, C10.10C42, C5×C2.C42, C2×C4×Dic5, C2×C4×Dic5, C2×C10.D4, Dic52C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C42, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, D10, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C4×D5, C22×D5, C4×C4⋊C4, C2×C4×D5, D4×D5, D42D5, Q8×D5, D5×C42, C23.11D10, Dic54D4, Dic53Q8, D5×C4⋊C4, Dic52C42

Smallest permutation representation of Dic52C42
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)
(1 267 6 262)(2 266 7 261)(3 265 8 270)(4 264 9 269)(5 263 10 268)(11 83 16 88)(12 82 17 87)(13 81 18 86)(14 90 19 85)(15 89 20 84)(21 257 26 252)(22 256 27 251)(23 255 28 260)(24 254 29 259)(25 253 30 258)(31 274 36 279)(32 273 37 278)(33 272 38 277)(34 271 39 276)(35 280 40 275)(41 289 46 284)(42 288 47 283)(43 287 48 282)(44 286 49 281)(45 285 50 290)(51 294 56 299)(52 293 57 298)(53 292 58 297)(54 291 59 296)(55 300 60 295)(61 309 66 304)(62 308 67 303)(63 307 68 302)(64 306 69 301)(65 305 70 310)(71 314 76 319)(72 313 77 318)(73 312 78 317)(74 311 79 316)(75 320 80 315)(91 190 96 185)(92 189 97 184)(93 188 98 183)(94 187 99 182)(95 186 100 181)(101 173 106 178)(102 172 107 177)(103 171 108 176)(104 180 109 175)(105 179 110 174)(111 210 116 205)(112 209 117 204)(113 208 118 203)(114 207 119 202)(115 206 120 201)(121 193 126 198)(122 192 127 197)(123 191 128 196)(124 200 129 195)(125 199 130 194)(131 230 136 225)(132 229 137 224)(133 228 138 223)(134 227 139 222)(135 226 140 221)(141 213 146 218)(142 212 147 217)(143 211 148 216)(144 220 149 215)(145 219 150 214)(151 250 156 245)(152 249 157 244)(153 248 158 243)(154 247 159 242)(155 246 160 241)(161 240 166 235)(162 239 167 234)(163 238 168 233)(164 237 169 232)(165 236 170 231)
(1 187 22 176)(2 186 23 175)(3 185 24 174)(4 184 25 173)(5 183 26 172)(6 182 27 171)(7 181 28 180)(8 190 29 179)(9 189 30 178)(10 188 21 177)(11 154 311 165)(12 153 312 164)(13 152 313 163)(14 151 314 162)(15 160 315 161)(16 159 316 170)(17 158 317 169)(18 157 318 168)(19 156 319 167)(20 155 320 166)(31 194 45 205)(32 193 46 204)(33 192 47 203)(34 191 48 202)(35 200 49 201)(36 199 50 210)(37 198 41 209)(38 197 42 208)(39 196 43 207)(40 195 44 206)(51 214 65 225)(52 213 66 224)(53 212 67 223)(54 211 68 222)(55 220 69 221)(56 219 70 230)(57 218 61 229)(58 217 62 228)(59 216 63 227)(60 215 64 226)(71 234 85 245)(72 233 86 244)(73 232 87 243)(74 231 88 242)(75 240 89 241)(76 239 90 250)(77 238 81 249)(78 237 82 248)(79 236 83 247)(80 235 84 246)(91 254 105 265)(92 253 106 264)(93 252 107 263)(94 251 108 262)(95 260 109 261)(96 259 110 270)(97 258 101 269)(98 257 102 268)(99 256 103 267)(100 255 104 266)(111 274 125 285)(112 273 126 284)(113 272 127 283)(114 271 128 282)(115 280 129 281)(116 279 130 290)(117 278 121 289)(118 277 122 288)(119 276 123 287)(120 275 124 286)(131 294 145 305)(132 293 146 304)(133 292 147 303)(134 291 148 302)(135 300 149 301)(136 299 150 310)(137 298 141 309)(138 297 142 308)(139 296 143 307)(140 295 144 306)
(1 83 43 63)(2 84 44 64)(3 85 45 65)(4 86 46 66)(5 87 47 67)(6 88 48 68)(7 89 49 69)(8 90 50 70)(9 81 41 61)(10 82 42 62)(11 287 302 267)(12 288 303 268)(13 289 304 269)(14 290 305 270)(15 281 306 261)(16 282 307 262)(17 283 308 263)(18 284 309 264)(19 285 310 265)(20 286 301 266)(21 78 38 58)(22 79 39 59)(23 80 40 60)(24 71 31 51)(25 72 32 52)(26 73 33 53)(27 74 34 54)(28 75 35 55)(29 76 36 56)(30 77 37 57)(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)(101 163 121 146)(102 164 122 147)(103 165 123 148)(104 166 124 149)(105 167 125 150)(106 168 126 141)(107 169 127 142)(108 170 128 143)(109 161 129 144)(110 162 130 145)(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)(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), (1,267,6,262)(2,266,7,261)(3,265,8,270)(4,264,9,269)(5,263,10,268)(11,83,16,88)(12,82,17,87)(13,81,18,86)(14,90,19,85)(15,89,20,84)(21,257,26,252)(22,256,27,251)(23,255,28,260)(24,254,29,259)(25,253,30,258)(31,274,36,279)(32,273,37,278)(33,272,38,277)(34,271,39,276)(35,280,40,275)(41,289,46,284)(42,288,47,283)(43,287,48,282)(44,286,49,281)(45,285,50,290)(51,294,56,299)(52,293,57,298)(53,292,58,297)(54,291,59,296)(55,300,60,295)(61,309,66,304)(62,308,67,303)(63,307,68,302)(64,306,69,301)(65,305,70,310)(71,314,76,319)(72,313,77,318)(73,312,78,317)(74,311,79,316)(75,320,80,315)(91,190,96,185)(92,189,97,184)(93,188,98,183)(94,187,99,182)(95,186,100,181)(101,173,106,178)(102,172,107,177)(103,171,108,176)(104,180,109,175)(105,179,110,174)(111,210,116,205)(112,209,117,204)(113,208,118,203)(114,207,119,202)(115,206,120,201)(121,193,126,198)(122,192,127,197)(123,191,128,196)(124,200,129,195)(125,199,130,194)(131,230,136,225)(132,229,137,224)(133,228,138,223)(134,227,139,222)(135,226,140,221)(141,213,146,218)(142,212,147,217)(143,211,148,216)(144,220,149,215)(145,219,150,214)(151,250,156,245)(152,249,157,244)(153,248,158,243)(154,247,159,242)(155,246,160,241)(161,240,166,235)(162,239,167,234)(163,238,168,233)(164,237,169,232)(165,236,170,231), (1,187,22,176)(2,186,23,175)(3,185,24,174)(4,184,25,173)(5,183,26,172)(6,182,27,171)(7,181,28,180)(8,190,29,179)(9,189,30,178)(10,188,21,177)(11,154,311,165)(12,153,312,164)(13,152,313,163)(14,151,314,162)(15,160,315,161)(16,159,316,170)(17,158,317,169)(18,157,318,168)(19,156,319,167)(20,155,320,166)(31,194,45,205)(32,193,46,204)(33,192,47,203)(34,191,48,202)(35,200,49,201)(36,199,50,210)(37,198,41,209)(38,197,42,208)(39,196,43,207)(40,195,44,206)(51,214,65,225)(52,213,66,224)(53,212,67,223)(54,211,68,222)(55,220,69,221)(56,219,70,230)(57,218,61,229)(58,217,62,228)(59,216,63,227)(60,215,64,226)(71,234,85,245)(72,233,86,244)(73,232,87,243)(74,231,88,242)(75,240,89,241)(76,239,90,250)(77,238,81,249)(78,237,82,248)(79,236,83,247)(80,235,84,246)(91,254,105,265)(92,253,106,264)(93,252,107,263)(94,251,108,262)(95,260,109,261)(96,259,110,270)(97,258,101,269)(98,257,102,268)(99,256,103,267)(100,255,104,266)(111,274,125,285)(112,273,126,284)(113,272,127,283)(114,271,128,282)(115,280,129,281)(116,279,130,290)(117,278,121,289)(118,277,122,288)(119,276,123,287)(120,275,124,286)(131,294,145,305)(132,293,146,304)(133,292,147,303)(134,291,148,302)(135,300,149,301)(136,299,150,310)(137,298,141,309)(138,297,142,308)(139,296,143,307)(140,295,144,306), (1,83,43,63)(2,84,44,64)(3,85,45,65)(4,86,46,66)(5,87,47,67)(6,88,48,68)(7,89,49,69)(8,90,50,70)(9,81,41,61)(10,82,42,62)(11,287,302,267)(12,288,303,268)(13,289,304,269)(14,290,305,270)(15,281,306,261)(16,282,307,262)(17,283,308,263)(18,284,309,264)(19,285,310,265)(20,286,301,266)(21,78,38,58)(22,79,39,59)(23,80,40,60)(24,71,31,51)(25,72,32,52)(26,73,33,53)(27,74,34,54)(28,75,35,55)(29,76,36,56)(30,77,37,57)(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)(101,163,121,146)(102,164,122,147)(103,165,123,148)(104,166,124,149)(105,167,125,150)(106,168,126,141)(107,169,127,142)(108,170,128,143)(109,161,129,144)(110,162,130,145)(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)(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), (1,267,6,262)(2,266,7,261)(3,265,8,270)(4,264,9,269)(5,263,10,268)(11,83,16,88)(12,82,17,87)(13,81,18,86)(14,90,19,85)(15,89,20,84)(21,257,26,252)(22,256,27,251)(23,255,28,260)(24,254,29,259)(25,253,30,258)(31,274,36,279)(32,273,37,278)(33,272,38,277)(34,271,39,276)(35,280,40,275)(41,289,46,284)(42,288,47,283)(43,287,48,282)(44,286,49,281)(45,285,50,290)(51,294,56,299)(52,293,57,298)(53,292,58,297)(54,291,59,296)(55,300,60,295)(61,309,66,304)(62,308,67,303)(63,307,68,302)(64,306,69,301)(65,305,70,310)(71,314,76,319)(72,313,77,318)(73,312,78,317)(74,311,79,316)(75,320,80,315)(91,190,96,185)(92,189,97,184)(93,188,98,183)(94,187,99,182)(95,186,100,181)(101,173,106,178)(102,172,107,177)(103,171,108,176)(104,180,109,175)(105,179,110,174)(111,210,116,205)(112,209,117,204)(113,208,118,203)(114,207,119,202)(115,206,120,201)(121,193,126,198)(122,192,127,197)(123,191,128,196)(124,200,129,195)(125,199,130,194)(131,230,136,225)(132,229,137,224)(133,228,138,223)(134,227,139,222)(135,226,140,221)(141,213,146,218)(142,212,147,217)(143,211,148,216)(144,220,149,215)(145,219,150,214)(151,250,156,245)(152,249,157,244)(153,248,158,243)(154,247,159,242)(155,246,160,241)(161,240,166,235)(162,239,167,234)(163,238,168,233)(164,237,169,232)(165,236,170,231), (1,187,22,176)(2,186,23,175)(3,185,24,174)(4,184,25,173)(5,183,26,172)(6,182,27,171)(7,181,28,180)(8,190,29,179)(9,189,30,178)(10,188,21,177)(11,154,311,165)(12,153,312,164)(13,152,313,163)(14,151,314,162)(15,160,315,161)(16,159,316,170)(17,158,317,169)(18,157,318,168)(19,156,319,167)(20,155,320,166)(31,194,45,205)(32,193,46,204)(33,192,47,203)(34,191,48,202)(35,200,49,201)(36,199,50,210)(37,198,41,209)(38,197,42,208)(39,196,43,207)(40,195,44,206)(51,214,65,225)(52,213,66,224)(53,212,67,223)(54,211,68,222)(55,220,69,221)(56,219,70,230)(57,218,61,229)(58,217,62,228)(59,216,63,227)(60,215,64,226)(71,234,85,245)(72,233,86,244)(73,232,87,243)(74,231,88,242)(75,240,89,241)(76,239,90,250)(77,238,81,249)(78,237,82,248)(79,236,83,247)(80,235,84,246)(91,254,105,265)(92,253,106,264)(93,252,107,263)(94,251,108,262)(95,260,109,261)(96,259,110,270)(97,258,101,269)(98,257,102,268)(99,256,103,267)(100,255,104,266)(111,274,125,285)(112,273,126,284)(113,272,127,283)(114,271,128,282)(115,280,129,281)(116,279,130,290)(117,278,121,289)(118,277,122,288)(119,276,123,287)(120,275,124,286)(131,294,145,305)(132,293,146,304)(133,292,147,303)(134,291,148,302)(135,300,149,301)(136,299,150,310)(137,298,141,309)(138,297,142,308)(139,296,143,307)(140,295,144,306), (1,83,43,63)(2,84,44,64)(3,85,45,65)(4,86,46,66)(5,87,47,67)(6,88,48,68)(7,89,49,69)(8,90,50,70)(9,81,41,61)(10,82,42,62)(11,287,302,267)(12,288,303,268)(13,289,304,269)(14,290,305,270)(15,281,306,261)(16,282,307,262)(17,283,308,263)(18,284,309,264)(19,285,310,265)(20,286,301,266)(21,78,38,58)(22,79,39,59)(23,80,40,60)(24,71,31,51)(25,72,32,52)(26,73,33,53)(27,74,34,54)(28,75,35,55)(29,76,36,56)(30,77,37,57)(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)(101,163,121,146)(102,164,122,147)(103,165,123,148)(104,166,124,149)(105,167,125,150)(106,168,126,141)(107,169,127,142)(108,170,128,143)(109,161,129,144)(110,162,130,145)(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)(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)], [(1,267,6,262),(2,266,7,261),(3,265,8,270),(4,264,9,269),(5,263,10,268),(11,83,16,88),(12,82,17,87),(13,81,18,86),(14,90,19,85),(15,89,20,84),(21,257,26,252),(22,256,27,251),(23,255,28,260),(24,254,29,259),(25,253,30,258),(31,274,36,279),(32,273,37,278),(33,272,38,277),(34,271,39,276),(35,280,40,275),(41,289,46,284),(42,288,47,283),(43,287,48,282),(44,286,49,281),(45,285,50,290),(51,294,56,299),(52,293,57,298),(53,292,58,297),(54,291,59,296),(55,300,60,295),(61,309,66,304),(62,308,67,303),(63,307,68,302),(64,306,69,301),(65,305,70,310),(71,314,76,319),(72,313,77,318),(73,312,78,317),(74,311,79,316),(75,320,80,315),(91,190,96,185),(92,189,97,184),(93,188,98,183),(94,187,99,182),(95,186,100,181),(101,173,106,178),(102,172,107,177),(103,171,108,176),(104,180,109,175),(105,179,110,174),(111,210,116,205),(112,209,117,204),(113,208,118,203),(114,207,119,202),(115,206,120,201),(121,193,126,198),(122,192,127,197),(123,191,128,196),(124,200,129,195),(125,199,130,194),(131,230,136,225),(132,229,137,224),(133,228,138,223),(134,227,139,222),(135,226,140,221),(141,213,146,218),(142,212,147,217),(143,211,148,216),(144,220,149,215),(145,219,150,214),(151,250,156,245),(152,249,157,244),(153,248,158,243),(154,247,159,242),(155,246,160,241),(161,240,166,235),(162,239,167,234),(163,238,168,233),(164,237,169,232),(165,236,170,231)], [(1,187,22,176),(2,186,23,175),(3,185,24,174),(4,184,25,173),(5,183,26,172),(6,182,27,171),(7,181,28,180),(8,190,29,179),(9,189,30,178),(10,188,21,177),(11,154,311,165),(12,153,312,164),(13,152,313,163),(14,151,314,162),(15,160,315,161),(16,159,316,170),(17,158,317,169),(18,157,318,168),(19,156,319,167),(20,155,320,166),(31,194,45,205),(32,193,46,204),(33,192,47,203),(34,191,48,202),(35,200,49,201),(36,199,50,210),(37,198,41,209),(38,197,42,208),(39,196,43,207),(40,195,44,206),(51,214,65,225),(52,213,66,224),(53,212,67,223),(54,211,68,222),(55,220,69,221),(56,219,70,230),(57,218,61,229),(58,217,62,228),(59,216,63,227),(60,215,64,226),(71,234,85,245),(72,233,86,244),(73,232,87,243),(74,231,88,242),(75,240,89,241),(76,239,90,250),(77,238,81,249),(78,237,82,248),(79,236,83,247),(80,235,84,246),(91,254,105,265),(92,253,106,264),(93,252,107,263),(94,251,108,262),(95,260,109,261),(96,259,110,270),(97,258,101,269),(98,257,102,268),(99,256,103,267),(100,255,104,266),(111,274,125,285),(112,273,126,284),(113,272,127,283),(114,271,128,282),(115,280,129,281),(116,279,130,290),(117,278,121,289),(118,277,122,288),(119,276,123,287),(120,275,124,286),(131,294,145,305),(132,293,146,304),(133,292,147,303),(134,291,148,302),(135,300,149,301),(136,299,150,310),(137,298,141,309),(138,297,142,308),(139,296,143,307),(140,295,144,306)], [(1,83,43,63),(2,84,44,64),(3,85,45,65),(4,86,46,66),(5,87,47,67),(6,88,48,68),(7,89,49,69),(8,90,50,70),(9,81,41,61),(10,82,42,62),(11,287,302,267),(12,288,303,268),(13,289,304,269),(14,290,305,270),(15,281,306,261),(16,282,307,262),(17,283,308,263),(18,284,309,264),(19,285,310,265),(20,286,301,266),(21,78,38,58),(22,79,39,59),(23,80,40,60),(24,71,31,51),(25,72,32,52),(26,73,33,53),(27,74,34,54),(28,75,35,55),(29,76,36,56),(30,77,37,57),(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),(101,163,121,146),(102,164,122,147),(103,165,123,148),(104,166,124,149),(105,167,125,150),(106,168,126,141),(107,169,127,142),(108,170,128,143),(109,161,129,144),(110,162,130,145),(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),(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 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + - + + + - - image C1 C2 C2 C2 C2 C4 C4 D4 Q8 D5 C4○D4 D10 C4×D5 D4×D5 D4⋊2D5 Q8×D5 kernel Dic5⋊2C42 C10.10C42 C5×C2.C42 C2×C4×Dic5 C2×C10.D4 C4×Dic5 C10.D4 C2×Dic5 C2×Dic5 C2.C42 C2×C10 C22×C4 C2×C4 C22 C22 C22 # reps 1 1 1 3 2 8 16 2 2 2 4 6 24 2 4 2

Matrix representation of Dic52C42 in GL6(𝔽41)

 0 1 0 0 0 0 40 34 0 0 0 0 0 0 0 1 0 0 0 0 40 34 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 7 1 0 0 0 0 34 34 0 0 0 0 0 0 39 21 0 0 0 0 35 2 0 0 0 0 0 0 2 39 0 0 0 0 23 39
,
 34 40 0 0 0 0 7 7 0 0 0 0 0 0 23 25 0 0 0 0 28 18 0 0 0 0 0 0 32 0 0 0 0 0 0 32
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 18 0 0 0 0 0 1

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,34,0,0,0,0,0,0,0,40,0,0,0,0,1,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[7,34,0,0,0,0,1,34,0,0,0,0,0,0,39,35,0,0,0,0,21,2,0,0,0,0,0,0,2,23,0,0,0,0,39,39],[34,7,0,0,0,0,40,7,0,0,0,0,0,0,23,28,0,0,0,0,25,18,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,18,1] >;

Dic52C42 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes_2C_4^2
% in TeX

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

G:=SmallGroup(320,276);
// by ID

G=gap.SmallGroup(320,276);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,555,58,12550]);
// Polycyclic

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

׿
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