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## G = C5⋊2(C42⋊5C4)  order 320 = 26·5

### The semidirect product of C5 and C42⋊5C4 acting via C42⋊5C4/C2.C42=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C5⋊2(C42⋊5C4)
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C2×C4×Dic5 — C5⋊2(C42⋊5C4)
 Lower central C5 — C2×C10 — C5⋊2(C42⋊5C4)
 Upper central C1 — C23 — C2.C42

Generators and relations for C52(C425C4)
G = < a,b,c,d | a5=b4=c4=d4=1, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c-1 >

Subgroups: 430 in 138 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C42, C22×C4, C22×C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2.C42, C2.C42, C2×C42, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C425C4, C4×Dic5, C22×Dic5, C22×Dic5, C22×C20, C22×C20, C10.10C42, C10.10C42, C5×C2.C42, C2×C4×Dic5, C52(C425C4)
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C4○D4, D10, C42⋊C2, C422C2, C4×D5, C22×D5, C425C4, C2×C4×D5, C4○D20, D42D5, Q82D5, C42⋊D5, C23.11D10, C23.D10, C4⋊C47D5, C4⋊C4⋊D5, C52(C425C4)

Smallest permutation representation of C52(C425C4)
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)
(1 276 19 261)(2 280 20 265)(3 279 16 264)(4 278 17 263)(5 277 18 262)(6 274 11 269)(7 273 12 268)(8 272 13 267)(9 271 14 266)(10 275 15 270)(21 259 36 244)(22 258 37 243)(23 257 38 242)(24 256 39 241)(25 260 40 245)(26 254 31 249)(27 253 32 248)(28 252 33 247)(29 251 34 246)(30 255 35 250)(41 319 56 304)(42 318 57 303)(43 317 58 302)(44 316 59 301)(45 320 60 305)(46 314 51 309)(47 313 52 308)(48 312 53 307)(49 311 54 306)(50 315 55 310)(61 299 76 284)(62 298 77 283)(63 297 78 282)(64 296 79 281)(65 300 80 285)(66 294 71 289)(67 293 72 288)(68 292 73 287)(69 291 74 286)(70 295 75 290)(81 189 96 194)(82 188 97 193)(83 187 98 192)(84 186 99 191)(85 190 100 195)(86 184 91 199)(87 183 92 198)(88 182 93 197)(89 181 94 196)(90 185 95 200)(101 169 116 174)(102 168 117 173)(103 167 118 172)(104 166 119 171)(105 170 120 175)(106 164 111 179)(107 163 112 178)(108 162 113 177)(109 161 114 176)(110 165 115 180)(121 229 136 234)(122 228 137 233)(123 227 138 232)(124 226 139 231)(125 230 140 235)(126 224 131 239)(127 223 132 238)(128 222 133 237)(129 221 134 236)(130 225 135 240)(141 209 156 214)(142 208 157 213)(143 207 158 212)(144 206 159 211)(145 210 160 215)(146 204 151 219)(147 203 152 218)(148 202 153 217)(149 201 154 216)(150 205 155 220)
(1 114 14 104)(2 115 15 105)(3 111 11 101)(4 112 12 102)(5 113 13 103)(6 116 16 106)(7 117 17 107)(8 118 18 108)(9 119 19 109)(10 120 20 110)(21 91 31 81)(22 92 32 82)(23 93 33 83)(24 94 34 84)(25 95 35 85)(26 96 36 86)(27 97 37 87)(28 98 38 88)(29 99 39 89)(30 100 40 90)(41 151 51 141)(42 152 52 142)(43 153 53 143)(44 154 54 144)(45 155 55 145)(46 156 56 146)(47 157 57 147)(48 158 58 148)(49 159 59 149)(50 160 60 150)(61 131 71 121)(62 132 72 122)(63 133 73 123)(64 134 74 124)(65 135 75 125)(66 136 76 126)(67 137 77 127)(68 138 78 128)(69 139 79 129)(70 140 80 130)(161 271 171 261)(162 272 172 262)(163 273 173 263)(164 274 174 264)(165 275 175 265)(166 276 176 266)(167 277 177 267)(168 278 178 268)(169 279 179 269)(170 280 180 270)(181 251 191 241)(182 252 192 242)(183 253 193 243)(184 254 194 244)(185 255 195 245)(186 256 196 246)(187 257 197 247)(188 258 198 248)(189 259 199 249)(190 260 200 250)(201 311 211 301)(202 312 212 302)(203 313 213 303)(204 314 214 304)(205 315 215 305)(206 316 216 306)(207 317 217 307)(208 318 218 308)(209 319 219 309)(210 320 220 310)(221 291 231 281)(222 292 232 282)(223 293 233 283)(224 294 234 284)(225 295 235 285)(226 296 236 286)(227 297 237 287)(228 298 238 288)(229 299 239 289)(230 300 240 290)
(1 64 24 44)(2 65 25 45)(3 61 21 41)(4 62 22 42)(5 63 23 43)(6 66 26 46)(7 67 27 47)(8 68 28 48)(9 69 29 49)(10 70 30 50)(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)(161 236 181 216)(162 237 182 217)(163 238 183 218)(164 239 184 219)(165 240 185 220)(166 231 186 211)(167 232 187 212)(168 233 188 213)(169 234 189 214)(170 235 190 215)(171 226 191 206)(172 227 192 207)(173 228 193 208)(174 229 194 209)(175 230 195 210)(176 221 196 201)(177 222 197 202)(178 223 198 203)(179 224 199 204)(180 225 200 205)(241 311 261 291)(242 312 262 292)(243 313 263 293)(244 314 264 294)(245 315 265 295)(246 316 266 296)(247 317 267 297)(248 318 268 298)(249 319 269 299)(250 320 270 300)(251 301 271 281)(252 302 272 282)(253 303 273 283)(254 304 274 284)(255 305 275 285)(256 306 276 286)(257 307 277 287)(258 308 278 288)(259 309 279 289)(260 310 280 290)

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

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

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

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 2 2 2 2 2 4 4 type + + + + + + - + image C1 C2 C2 C2 C4 D5 C4○D4 D10 C4×D5 C4○D20 D4⋊2D5 Q8⋊2D5 kernel C5⋊2(C42⋊5C4) C10.10C42 C5×C2.C42 C2×C4×Dic5 C4×Dic5 C2.C42 C2×C10 C22×C4 C2×C4 C22 C22 C22 # reps 1 5 1 1 8 2 12 6 8 16 6 2

Matrix representation of C52(C425C4) in GL5(𝔽41)

 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 40 1 0 0 0 33 7
,
 40 0 0 0 0 0 16 38 0 0 0 3 25 0 0 0 0 0 3 18 0 0 0 4 38
,
 1 0 0 0 0 0 20 27 0 0 0 14 21 0 0 0 0 0 9 0 0 0 0 0 9
,
 32 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 17 6 0 0 0 34 24

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,33,0,0,0,1,7],[40,0,0,0,0,0,16,3,0,0,0,38,25,0,0,0,0,0,3,4,0,0,0,18,38],[1,0,0,0,0,0,20,14,0,0,0,27,21,0,0,0,0,0,9,0,0,0,0,0,9],[32,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,17,34,0,0,0,6,24] >;

C52(C425C4) in GAP, Magma, Sage, TeX

C_5\rtimes_2(C_4^2\rtimes_5C_4)
% in TeX

G:=Group("C5:2(C4^2:5C4)");
// GroupNames label

G:=SmallGroup(320,278);
// by ID

G=gap.SmallGroup(320,278);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,1094,387,58,12550]);
// Polycyclic

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

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