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G = C10.52(C4×D4)  order 320 = 26·5

4th non-split extension by C10 of C4×D4 acting via C4×D4/C22⋊C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.52(C4×D4), (C2×C20).35Q8, C10.13(C4×Q8), C10.D46C4, C2.7(C4×Dic10), C22.56(D4×D5), (C2×C4).22Dic10, (C22×C4).66D10, C10.4(C22⋊Q8), (C2×Dic5).185D4, (C22×C20).6C22, C2.2(C4.Dic10), C10.9(C422C2), C2.7(Dic54D4), C22.32(C4○D20), C10.10(C42.C2), C22.15(C2×Dic10), C2.C42.11D5, C23.252(C22×D5), C10.47(C42⋊C2), C2.1(D10.13D4), C22.34(D42D5), C10.10C42.4C2, (C22×C10).280C23, C55(C23.63C23), C22.15(Q82D5), C2.4(C23.D10), C2.4(Dic5.14D4), C10.36(C22.D4), (C22×Dic5).197C22, (C2×C4).26(C4×D5), C22.87(C2×C4×D5), (C2×C4⋊Dic5).6C2, (C2×C10).20(C2×Q8), (C2×C4×Dic5).29C2, (C2×C20).207(C2×C4), C2.6(C4⋊C47D5), (C2×C10).193(C2×D4), (C2×C10).56(C4○D4), (C2×Dic5).10(C2×C4), (C2×C10).147(C22×C4), (C2×C10.D4).19C2, (C5×C2.C42).18C2, SmallGroup(320,282)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C10.52(C4×D4)
C1C5C10C2×C10C22×C10C22×Dic5C2×C4×Dic5 — C10.52(C4×D4)
C5C2×C10 — C10.52(C4×D4)
C1C23C2.C42

Generators and relations for C10.52(C4×D4)
 G = < a,b,c,d | a10=b4=c4=1, d2=a5, bab-1=cac-1=a-1, ad=da, cbc-1=dbd-1=a5b, dcd-1=c-1 >

Subgroups: 478 in 154 conjugacy classes, 69 normal (51 characteristic)
C1, C2 [×7], C4 [×12], C22 [×7], C5, C2×C4 [×4], C2×C4 [×22], C23, C10 [×7], C42 [×2], C4⋊C4 [×6], C22×C4 [×3], C22×C4 [×4], Dic5 [×7], C20 [×5], C2×C10 [×7], C2.C42, C2.C42 [×3], C2×C42, C2×C4⋊C4 [×2], C2×Dic5 [×6], C2×Dic5 [×9], C2×C20 [×4], C2×C20 [×7], C22×C10, C23.63C23, C4×Dic5 [×2], C10.D4 [×4], C4⋊Dic5 [×2], C22×Dic5 [×4], C22×C20 [×3], C10.10C42 [×3], C5×C2.C42, C2×C4×Dic5, C2×C10.D4, C2×C4⋊Dic5, C10.52(C4×D4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C22×C4, C2×D4, C2×Q8, C4○D4 [×4], D10 [×3], C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, Dic10 [×2], C4×D5 [×2], C22×D5, C23.63C23, C2×Dic10, C2×C4×D5, C4○D20, D4×D5, D42D5 [×2], Q82D5, C4×Dic10, Dic5.14D4, C23.D10, Dic54D4, C4.Dic10, C4⋊C47D5, D10.13D4, C10.52(C4×D4)

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

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

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

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

68 conjugacy classes

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

68 irreducible representations

dim111111122222222444
type+++++++-++-+-+
imageC1C2C2C2C2C2C4D4Q8D5C4○D4D10Dic10C4×D5C4○D20D4×D5D42D5Q82D5
kernelC10.52(C4×D4)C10.10C42C5×C2.C42C2×C4×Dic5C2×C10.D4C2×C4⋊Dic5C10.D4C2×Dic5C2×C20C2.C42C2×C10C22×C4C2×C4C2×C4C22C22C22C22
# reps131111822286888242

Matrix representation of C10.52(C4×D4) in GL6(𝔽41)

100000
010000
000700
0035600
00003434
000071
,
4000000
0400000
00184000
00382300
0000229
00001919
,
010000
4000000
002900
00273900
00003817
0000383
,
16140000
14250000
0040000
0004000
00003032
0000911

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,35,0,0,0,0,7,6,0,0,0,0,0,0,34,7,0,0,0,0,34,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,18,38,0,0,0,0,40,23,0,0,0,0,0,0,22,19,0,0,0,0,9,19],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,2,27,0,0,0,0,9,39,0,0,0,0,0,0,38,38,0,0,0,0,17,3],[16,14,0,0,0,0,14,25,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,30,9,0,0,0,0,32,11] >;

C10.52(C4×D4) in GAP, Magma, Sage, TeX

C_{10}._{52}(C_4\times D_4)
% in TeX

G:=Group("C10.52(C4xD4)");
// GroupNames label

G:=SmallGroup(320,282);
// by ID

G=gap.SmallGroup(320,282);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,64,926,219,268,12550]);
// Polycyclic

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

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