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G = C52(C428C4)  order 320 = 26·5

The semidirect product of C5 and C428C4 acting via C428C4/C2.C42=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52(C428C4), (C4×Dic5)⋊15C4, C22.53(D4×D5), (C22×C4).7D10, C22.12(Q8×D5), (C2×Dic5).16Q8, Dic5.12(C4⋊C4), (C2×Dic5).128D4, C2.5(C42⋊D5), (C22×C20).3C22, C10.6(C42.C2), C2.C42.1D5, C10.15(C4.4D4), C22.27(C4○D20), C23.247(C22×D5), C10.20(C42⋊C2), C22.29(D42D5), C10.10C42.1C2, (C22×C10).275C23, C2.2(Dic5.Q8), C2.2(Dic5.5D4), (C22×Dic5).3C22, C2.5(C23.11D10), C2.6(D5×C4⋊C4), C10.26(C2×C4⋊C4), C22.82(C2×C4×D5), (C2×C4).122(C4×D5), (C2×C10).59(C2×Q8), (C2×C4×Dic5).24C2, (C2×C20).312(C2×C4), (C2×C10).190(C2×D4), (C2×C10).54(C4○D4), (C2×C10.D4).5C2, (C2×C10).142(C22×C4), (C2×Dic5).130(C2×C4), (C5×C2.C42).21C2, SmallGroup(320,277)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C52(C428C4)
C1C5C10C2×C10C22×C10C22×Dic5C2×C4×Dic5 — C52(C428C4)
C5C2×C10 — C52(C428C4)
C1C23C2.C42

Generators and relations for C52(C428C4)
 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 >

Subgroups: 478 in 154 conjugacy classes, 71 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C22 [×3], C22 [×4], C5, C2×C4 [×2], C2×C4 [×24], C23, C10 [×3], C10 [×4], C42 [×4], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], Dic5 [×4], Dic5 [×4], C20 [×4], C2×C10 [×3], C2×C10 [×4], C2.C42, C2.C42 [×3], C2×C42, C2×C4⋊C4 [×2], C2×Dic5 [×8], C2×Dic5 [×8], C2×C20 [×2], C2×C20 [×8], C22×C10, C428C4, C4×Dic5 [×4], C10.D4 [×4], C22×Dic5 [×2], C22×Dic5 [×2], C22×C20, C22×C20 [×2], C10.10C42, C10.10C42 [×2], C5×C2.C42, C2×C4×Dic5, C2×C10.D4 [×2], C52(C428C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C4○D4 [×4], D10 [×3], C2×C4⋊C4, C42⋊C2 [×2], C4.4D4 [×2], C42.C2 [×2], C4×D5 [×2], C22×D5, C428C4, C2×C4×D5, C4○D20 [×2], D4×D5, D42D5 [×2], Q8×D5, C42⋊D5, C23.11D10, Dic5.5D4 [×2], Dic5.Q8 [×2], D5×C4⋊C4, C52(C428C4)

Smallest permutation representation of C52(C428C4)
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)
(1 186 9 181)(2 190 10 185)(3 189 6 184)(4 188 7 183)(5 187 8 182)(11 199 16 194)(12 198 17 193)(13 197 18 192)(14 196 19 191)(15 200 20 195)(21 169 26 164)(22 168 27 163)(23 167 28 162)(24 166 29 161)(25 170 30 165)(31 179 36 174)(32 178 37 173)(33 177 38 172)(34 176 39 171)(35 180 40 175)(41 229 46 224)(42 228 47 223)(43 227 48 222)(44 226 49 221)(45 230 50 225)(51 239 56 234)(52 238 57 233)(53 237 58 232)(54 236 59 231)(55 240 60 235)(61 209 66 204)(62 208 67 203)(63 207 68 202)(64 206 69 201)(65 210 70 205)(71 219 76 214)(72 218 77 213)(73 217 78 212)(74 216 79 211)(75 220 80 215)(81 269 86 264)(82 268 87 263)(83 267 88 262)(84 266 89 261)(85 270 90 265)(91 279 96 274)(92 278 97 273)(93 277 98 272)(94 276 99 271)(95 280 100 275)(101 249 106 244)(102 248 107 243)(103 247 108 242)(104 246 109 241)(105 250 110 245)(111 259 116 254)(112 258 117 253)(113 257 118 252)(114 256 119 251)(115 260 120 255)(121 309 126 304)(122 308 127 303)(123 307 128 302)(124 306 129 301)(125 310 130 305)(131 319 136 314)(132 318 137 313)(133 317 138 312)(134 316 139 311)(135 320 140 315)(141 289 146 284)(142 288 147 283)(143 287 148 282)(144 286 149 281)(145 290 150 285)(151 299 156 294)(152 298 157 293)(153 297 158 292)(154 296 159 291)(155 300 160 295)
(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)(161 231 181 211)(162 232 182 212)(163 233 183 213)(164 234 184 214)(165 235 185 215)(166 236 186 216)(167 237 187 217)(168 238 188 218)(169 239 189 219)(170 240 190 220)(171 221 191 201)(172 222 192 202)(173 223 193 203)(174 224 194 204)(175 225 195 205)(176 226 196 206)(177 227 197 207)(178 228 198 208)(179 229 199 209)(180 230 200 210)(241 316 261 296)(242 317 262 297)(243 318 263 298)(244 319 264 299)(245 320 265 300)(246 311 266 291)(247 312 267 292)(248 313 268 293)(249 314 269 294)(250 315 270 295)(251 306 271 286)(252 307 272 287)(253 308 273 288)(254 309 274 289)(255 310 275 290)(256 301 276 281)(257 302 277 282)(258 303 278 283)(259 304 279 284)(260 305 280 285)

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

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

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

68 conjugacy classes

class 1 2A···2G4A4B4C4D4E4F4G4H4I···4P4Q4R4S4T5A5B10A···10N20A···20X
order12···2444444444···444445510···1020···20
size11···12222444410···1020202020222···24···4

68 irreducible representations

dim1111112222222444
type++++++-+++--
imageC1C2C2C2C2C4D4Q8D5C4○D4D10C4×D5C4○D20D4×D5D42D5Q8×D5
kernelC52(C428C4)C10.10C42C5×C2.C42C2×C4×Dic5C2×C10.D4C4×Dic5C2×Dic5C2×Dic5C2.C42C2×C10C22×C4C2×C4C22C22C22C22
# reps13112822286816242

Matrix representation of C52(C428C4) in GL6(𝔽41)

010000
4060000
001000
000100
000010
000001
,
660000
1350000
0032000
0003200
0000032
000090
,
100000
010000
0040000
000100
000090
000009
,
900000
090000
000100
0040000
000001
000010

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,1,0,0,0,0,6,35,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,9,0,0,0,0,32,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C52(C428C4) in GAP, Magma, Sage, TeX

C_5\rtimes_2(C_4^2\rtimes_8C_4)
% in TeX

G:=Group("C5:2(C4^2:8C4)");
// GroupNames label

G:=SmallGroup(320,277);
// by ID

G=gap.SmallGroup(320,277);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,422,387,58,12550]);
// 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>;
// generators/relations

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