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G = C42.71D10order 320 = 26·5

71st non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.71D10, C4⋊C4.78D10, (C2×C20).86D4, C42.C2.5D5, C20.73(C4○D4), C202Q8.18C2, (C4×C20).117C22, (C2×C20).387C23, C4.15(Q82D5), C10.Q16.13C2, C42.D5.6C2, C10.57(C4.4D4), C2.22(D4.9D10), C2.10(C20.23D4), C10.123(C8.C22), C53(C42.30C22), (C2×Dic10).114C22, (C2×C10).518(C2×D4), (C2×C4).68(C5⋊D4), (C5×C42.C2).4C2, (C5×C4⋊C4).125C22, (C2×C4).485(C22×D5), C22.191(C2×C5⋊D4), (C2×C52C8).128C22, SmallGroup(320,696)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C42.71D10
C1C5C10C20C2×C20C2×Dic10C202Q8 — C42.71D10
C5C10C2×C20 — C42.71D10
C1C22C42C42.C2

Generators and relations for C42.71D10
 G = < a,b,c,d | a4=b4=1, c10=a2, d2=a2b, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c9 >

Subgroups: 302 in 90 conjugacy classes, 39 normal (15 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×6], C22, C5, C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C10, C10 [×2], C42, C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C2×Q8 [×2], Dic5 [×2], C20 [×2], C20 [×4], C2×C10, C8⋊C4, Q8⋊C4 [×4], C42.C2, C4⋊Q8, C52C8 [×2], Dic10 [×4], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C2×C20 [×2], C42.30C22, C2×C52C8 [×2], C4⋊Dic5 [×2], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10 [×2], C42.D5, C10.Q16 [×4], C202Q8, C5×C42.C2, C42.71D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C8.C22 [×2], C5⋊D4 [×2], C22×D5, C42.30C22, Q82D5 [×2], C2×C5⋊D4, C20.23D4, D4.9D10 [×2], C42.71D10

Smallest permutation representation of C42.71D10
Regular action on 320 points
Generators in S320
(1 82 11 92)(2 235 12 225)(3 84 13 94)(4 237 14 227)(5 86 15 96)(6 239 16 229)(7 88 17 98)(8 221 18 231)(9 90 19 100)(10 223 20 233)(21 236 31 226)(22 85 32 95)(23 238 33 228)(24 87 34 97)(25 240 35 230)(26 89 36 99)(27 222 37 232)(28 91 38 81)(29 224 39 234)(30 93 40 83)(41 256 51 246)(42 288 52 298)(43 258 53 248)(44 290 54 300)(45 260 55 250)(46 292 56 282)(47 242 57 252)(48 294 58 284)(49 244 59 254)(50 296 60 286)(61 138 71 128)(62 279 72 269)(63 140 73 130)(64 261 74 271)(65 122 75 132)(66 263 76 273)(67 124 77 134)(68 265 78 275)(69 126 79 136)(70 267 80 277)(101 309 111 319)(102 153 112 143)(103 311 113 301)(104 155 114 145)(105 313 115 303)(106 157 116 147)(107 315 117 305)(108 159 118 149)(109 317 119 307)(110 141 120 151)(121 209 131 219)(123 211 133 201)(125 213 135 203)(127 215 137 205)(129 217 139 207)(142 166 152 176)(144 168 154 178)(146 170 156 180)(148 172 158 162)(150 174 160 164)(161 304 171 314)(163 306 173 316)(165 308 175 318)(167 310 177 320)(169 312 179 302)(181 247 191 257)(182 299 192 289)(183 249 193 259)(184 281 194 291)(185 251 195 241)(186 283 196 293)(187 253 197 243)(188 285 198 295)(189 255 199 245)(190 287 200 297)(202 264 212 274)(204 266 214 276)(206 268 216 278)(208 270 218 280)(210 272 220 262)
(1 277 29 127)(2 128 30 278)(3 279 31 129)(4 130 32 280)(5 261 33 131)(6 132 34 262)(7 263 35 133)(8 134 36 264)(9 265 37 135)(10 136 38 266)(11 267 39 137)(12 138 40 268)(13 269 21 139)(14 140 22 270)(15 271 23 121)(16 122 24 272)(17 273 25 123)(18 124 26 274)(19 275 27 125)(20 126 28 276)(41 156 200 303)(42 304 181 157)(43 158 182 305)(44 306 183 159)(45 160 184 307)(46 308 185 141)(47 142 186 309)(48 310 187 143)(49 144 188 311)(50 312 189 145)(51 146 190 313)(52 314 191 147)(53 148 192 315)(54 316 193 149)(55 150 194 317)(56 318 195 151)(57 152 196 319)(58 320 197 153)(59 154 198 301)(60 302 199 155)(61 93 206 235)(62 236 207 94)(63 95 208 237)(64 238 209 96)(65 97 210 239)(66 240 211 98)(67 99 212 221)(68 222 213 100)(69 81 214 223)(70 224 215 82)(71 83 216 225)(72 226 217 84)(73 85 218 227)(74 228 219 86)(75 87 220 229)(76 230 201 88)(77 89 202 231)(78 232 203 90)(79 91 204 233)(80 234 205 92)(101 252 176 293)(102 294 177 253)(103 254 178 295)(104 296 179 255)(105 256 180 297)(106 298 161 257)(107 258 162 299)(108 300 163 259)(109 260 164 281)(110 282 165 241)(111 242 166 283)(112 284 167 243)(113 244 168 285)(114 286 169 245)(115 246 170 287)(116 288 171 247)(117 248 172 289)(118 290 173 249)(119 250 174 291)(120 292 175 251)
(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)
(1 308 267 195 29 141 137 56)(2 194 138 307 30 55 268 160)(3 306 269 193 31 159 139 54)(4 192 140 305 32 53 270 158)(5 304 271 191 33 157 121 52)(6 190 122 303 34 51 272 156)(7 302 273 189 35 155 123 50)(8 188 124 301 36 49 274 154)(9 320 275 187 37 153 125 48)(10 186 126 319 38 47 276 152)(11 318 277 185 39 151 127 46)(12 184 128 317 40 45 278 150)(13 316 279 183 21 149 129 44)(14 182 130 315 22 43 280 148)(15 314 261 181 23 147 131 42)(16 200 132 313 24 41 262 146)(17 312 263 199 25 145 133 60)(18 198 134 311 26 59 264 144)(19 310 265 197 27 143 135 58)(20 196 136 309 28 57 266 142)(61 174 83 281 206 119 225 260)(62 300 226 173 207 259 84 118)(63 172 85 299 208 117 227 258)(64 298 228 171 209 257 86 116)(65 170 87 297 210 115 229 256)(66 296 230 169 211 255 88 114)(67 168 89 295 212 113 231 254)(68 294 232 167 213 253 90 112)(69 166 91 293 214 111 233 252)(70 292 234 165 215 251 92 110)(71 164 93 291 216 109 235 250)(72 290 236 163 217 249 94 108)(73 162 95 289 218 107 237 248)(74 288 238 161 219 247 96 106)(75 180 97 287 220 105 239 246)(76 286 240 179 201 245 98 104)(77 178 99 285 202 103 221 244)(78 284 222 177 203 243 100 102)(79 176 81 283 204 101 223 242)(80 282 224 175 205 241 82 120)

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

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

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

44 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F20A···20L20M···20T
order12224444444455888810···1020···2020···20
size1111224488404022202020202···24···48···8

44 irreducible representations

dim11111222222444
type+++++++++-+-
imageC1C2C2C2C2D4D5C4○D4D10D10C5⋊D4C8.C22Q82D5D4.9D10
kernelC42.71D10C42.D5C10.Q16C202Q8C5×C42.C2C2×C20C42.C2C20C42C4⋊C4C2×C4C10C4C2
# reps11411224248248

Matrix representation of C42.71D10 in GL6(𝔽41)

26230000
8150000
0000119
00003230
00303200
0091100
,
4000000
0400000
000010
000001
0040000
0004000
,
2590000
17160000
00622402
0019163926
004023519
0039262225
,
2920000
31120000
0016252321
0036252418
0018201625
0017233625

G:=sub<GL(6,GF(41))| [26,8,0,0,0,0,23,15,0,0,0,0,0,0,0,0,30,9,0,0,0,0,32,11,0,0,11,32,0,0,0,0,9,30,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0],[25,17,0,0,0,0,9,16,0,0,0,0,0,0,6,19,40,39,0,0,22,16,2,26,0,0,40,39,35,22,0,0,2,26,19,25],[29,31,0,0,0,0,2,12,0,0,0,0,0,0,16,36,18,17,0,0,25,25,20,23,0,0,23,24,16,36,0,0,21,18,25,25] >;

C42.71D10 in GAP, Magma, Sage, TeX

C_4^2._{71}D_{10}
% in TeX

G:=Group("C4^2.71D10");
// GroupNames label

G:=SmallGroup(320,696);
// by ID

G=gap.SmallGroup(320,696);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,120,254,555,100,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2,d^2=a^2*b,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^9>;
// generators/relations

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