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G = C42.59D10order 320 = 26·5

59th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.59D10, C5⋊Q169C4, (C4×Q8).7D5, Q8.6(C4×D5), (Q8×C20).8C2, C4⋊C4.256D10, C55(Q16⋊C4), (C2×C20).260D4, C10.106(C4×D4), C4.43(C4○D20), C20.63(C4○D4), C20.63(C22×C4), (C2×Q8).163D10, (C2×C20).350C23, (C4×C20).101C22, (C4×Dic10).15C2, Dic10.32(C2×C4), Q8⋊Dic5.10C2, C10.Q16.10C2, C20.Q8.12C2, C42.D5.4C2, C2.4(D4.9D10), C2.4(C20.C23), C4⋊Dic5.333C22, (Q8×C10).198C22, C10.111(C8.C22), (C2×Dic10).275C22, C4.28(C2×C4×D5), C52C8.4(C2×C4), C2.22(C4×C5⋊D4), (C5×Q8).28(C2×C4), (C2×C5⋊Q16).4C2, (C2×C10).481(C2×D4), C22.82(C2×C5⋊D4), (C2×C4).223(C5⋊D4), (C5×C4⋊C4).287C22, (C2×C4).450(C22×D5), (C2×C52C8).103C22, SmallGroup(320,657)

Series: Derived Chief Lower central Upper central

C1C20 — C42.59D10
C1C5C10C2×C10C2×C20C2×Dic10C2×C5⋊Q16 — C42.59D10
C5C10C20 — C42.59D10
C1C22C42C4×Q8

Generators and relations for C42.59D10
 G = < a,b,c,d | a4=b4=1, c10=b2, d2=cbc-1=b-1, ab=ba, cac-1=dad-1=ab2, bd=db, dcd-1=b-1c9 >

Subgroups: 310 in 108 conjugacy classes, 51 normal (39 characteristic)
C1, C2 [×3], C4 [×2], C4 [×8], C22, C5, C8 [×3], C2×C4 [×3], C2×C4 [×4], Q8 [×2], Q8 [×4], C10 [×3], C42, C42 [×2], C4⋊C4, C4⋊C4 [×3], C2×C8 [×2], Q16 [×4], C2×Q8, C2×Q8, Dic5 [×3], C20 [×2], C20 [×5], C2×C10, C8⋊C4, Q8⋊C4 [×2], C4.Q8, C4×Q8, C4×Q8, C2×Q16, C52C8 [×2], C52C8, Dic10 [×2], Dic10, C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×2], C5×Q8 [×2], C5×Q8, Q16⋊C4, C2×C52C8 [×2], C4×Dic5, C10.D4, C4⋊Dic5, C5⋊Q16 [×4], C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, Q8×C10, C42.D5, C20.Q8, C10.Q16, Q8⋊Dic5, C4×Dic10, C2×C5⋊Q16, Q8×C20, C42.59D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C8.C22 [×2], C4×D5 [×2], C5⋊D4 [×2], C22×D5, Q16⋊C4, C2×C4×D5, C4○D20, C2×C5⋊D4, C4×C5⋊D4, C20.C23, D4.9D10, C42.59D10

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

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

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

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

62 conjugacy classes

class 1 2A2B2C4A···4F4G4H4I4J4K4L4M4N5A5B8A8B8C8D10A···10F20A···20H20I···20AF
order12224···44444444455888810···1020···2020···20
size11112···244442020202022202020202···22···24···4

62 irreducible representations

dim111111111222222222444
type+++++++++++++--
imageC1C2C2C2C2C2C2C2C4D4D5C4○D4D10D10D10C5⋊D4C4×D5C4○D20C8.C22C20.C23D4.9D10
kernelC42.59D10C42.D5C20.Q8C10.Q16Q8⋊Dic5C4×Dic10C2×C5⋊Q16Q8×C20C5⋊Q16C2×C20C4×Q8C20C42C4⋊C4C2×Q8C2×C4Q8C4C10C2C2
# reps111111118222222888244

Matrix representation of C42.59D10 in GL6(𝔽41)

3200000
0320000
0000236
00003518
00183500
0062300
,
4000000
0400000
000010
000001
0040000
0004000
,
3100000
20370000
0035102911
0031133013
002911631
0030131028
,
35310000
1660000
001541834
0012261923
00237154
0022181226

G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,18,6,0,0,0,0,35,23,0,0,23,35,0,0,0,0,6,18,0,0],[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],[31,20,0,0,0,0,0,37,0,0,0,0,0,0,35,31,29,30,0,0,10,13,11,13,0,0,29,30,6,10,0,0,11,13,31,28],[35,16,0,0,0,0,31,6,0,0,0,0,0,0,15,12,23,22,0,0,4,26,7,18,0,0,18,19,15,12,0,0,34,23,4,26] >;

C42.59D10 in GAP, Magma, Sage, TeX

C_4^2._{59}D_{10}
% in TeX

G:=Group("C4^2.59D10");
// GroupNames label

G:=SmallGroup(320,657);
// by ID

G=gap.SmallGroup(320,657);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,232,387,58,1684,851,102,12550]);
// Polycyclic

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

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