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G = C40.26D4order 320 = 26·5

26th non-split extension by C40 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.26D4, Dic51Q16, C4.27(D4×D5), C20.51(C2×D4), C53(C4⋊Q16), C52C8.33D4, (C2×Q16).3D5, C2.16(D5×Q16), (C2×C8).241D10, C8.18(C5⋊D4), (C2×Q8).60D10, (C10×Q16).4C2, C10.26(C2×Q16), (C8×Dic5).4C2, (C2×C40).93C22, C22.274(D4×D5), C2.24(C20⋊D4), C10.33(C41D4), (C2×C20).455C23, (C2×Dic20).11C2, Dic5⋊Q8.7C2, (C2×Dic5).161D4, (Q8×C10).84C22, (C4×Dic5).274C22, (C2×Dic10).134C22, C4.14(C2×C5⋊D4), (C2×C5⋊Q16).8C2, (C2×C10).366(C2×D4), (C2×C4).543(C22×D5), (C2×C52C8).284C22, SmallGroup(320,808)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C40.26D4
C1C5C10C20C2×C20C4×Dic5Dic5⋊Q8 — C40.26D4
C5C10C2×C20 — C40.26D4
C1C22C2×C4C2×Q16

Generators and relations for C40.26D4
 G = < a,b,c | a40=b4=1, c2=a20, bab-1=a9, cac-1=a-1, cbc-1=b-1 >

Subgroups: 430 in 122 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×8], C22, C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×6], Q8 [×8], C10, C10 [×2], C42, C4⋊C4 [×4], C2×C8, C2×C8, Q16 [×8], C2×Q8 [×2], C2×Q8 [×2], Dic5 [×4], Dic5 [×2], C20 [×2], C20 [×2], C2×C10, C4×C8, C4⋊Q8 [×2], C2×Q16, C2×Q16 [×3], C52C8 [×2], C40 [×2], Dic10 [×4], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×4], C4⋊Q16, Dic20 [×2], C2×C52C8, C4×Dic5, C10.D4 [×4], C5⋊Q16 [×4], C2×C40, C5×Q16 [×2], C2×Dic10 [×2], Q8×C10 [×2], C8×Dic5, C2×Dic20, C2×C5⋊Q16 [×2], Dic5⋊Q8 [×2], C10×Q16, C40.26D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, Q16 [×4], C2×D4 [×3], D10 [×3], C41D4, C2×Q16 [×2], C5⋊D4 [×2], C22×D5, C4⋊Q16, D4×D5 [×2], C2×C5⋊D4, D5×Q16 [×2], C20⋊D4, C40.26D4

Smallest permutation representation of C40.26D4
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)
(1 252 218 300)(2 261 219 309)(3 270 220 318)(4 279 221 287)(5 248 222 296)(6 257 223 305)(7 266 224 314)(8 275 225 283)(9 244 226 292)(10 253 227 301)(11 262 228 310)(12 271 229 319)(13 280 230 288)(14 249 231 297)(15 258 232 306)(16 267 233 315)(17 276 234 284)(18 245 235 293)(19 254 236 302)(20 263 237 311)(21 272 238 320)(22 241 239 289)(23 250 240 298)(24 259 201 307)(25 268 202 316)(26 277 203 285)(27 246 204 294)(28 255 205 303)(29 264 206 312)(30 273 207 281)(31 242 208 290)(32 251 209 299)(33 260 210 308)(34 269 211 317)(35 278 212 286)(36 247 213 295)(37 256 214 304)(38 265 215 313)(39 274 216 282)(40 243 217 291)(41 86 132 176)(42 95 133 185)(43 104 134 194)(44 113 135 163)(45 82 136 172)(46 91 137 181)(47 100 138 190)(48 109 139 199)(49 118 140 168)(50 87 141 177)(51 96 142 186)(52 105 143 195)(53 114 144 164)(54 83 145 173)(55 92 146 182)(56 101 147 191)(57 110 148 200)(58 119 149 169)(59 88 150 178)(60 97 151 187)(61 106 152 196)(62 115 153 165)(63 84 154 174)(64 93 155 183)(65 102 156 192)(66 111 157 161)(67 120 158 170)(68 89 159 179)(69 98 160 188)(70 107 121 197)(71 116 122 166)(72 85 123 175)(73 94 124 184)(74 103 125 193)(75 112 126 162)(76 81 127 171)(77 90 128 180)(78 99 129 189)(79 108 130 198)(80 117 131 167)
(1 153 21 133)(2 152 22 132)(3 151 23 131)(4 150 24 130)(5 149 25 129)(6 148 26 128)(7 147 27 127)(8 146 28 126)(9 145 29 125)(10 144 30 124)(11 143 31 123)(12 142 32 122)(13 141 33 121)(14 140 34 160)(15 139 35 159)(16 138 36 158)(17 137 37 157)(18 136 38 156)(19 135 39 155)(20 134 40 154)(41 219 61 239)(42 218 62 238)(43 217 63 237)(44 216 64 236)(45 215 65 235)(46 214 66 234)(47 213 67 233)(48 212 68 232)(49 211 69 231)(50 210 70 230)(51 209 71 229)(52 208 72 228)(53 207 73 227)(54 206 74 226)(55 205 75 225)(56 204 76 224)(57 203 77 223)(58 202 78 222)(59 201 79 221)(60 240 80 220)(81 266 101 246)(82 265 102 245)(83 264 103 244)(84 263 104 243)(85 262 105 242)(86 261 106 241)(87 260 107 280)(88 259 108 279)(89 258 109 278)(90 257 110 277)(91 256 111 276)(92 255 112 275)(93 254 113 274)(94 253 114 273)(95 252 115 272)(96 251 116 271)(97 250 117 270)(98 249 118 269)(99 248 119 268)(100 247 120 267)(161 284 181 304)(162 283 182 303)(163 282 183 302)(164 281 184 301)(165 320 185 300)(166 319 186 299)(167 318 187 298)(168 317 188 297)(169 316 189 296)(170 315 190 295)(171 314 191 294)(172 313 192 293)(173 312 193 292)(174 311 194 291)(175 310 195 290)(176 309 196 289)(177 308 197 288)(178 307 198 287)(179 306 199 286)(180 305 200 285)

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

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

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

50 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D8E8F8G8H10A···10F20A20B20C20D20E···20L40A···40H
order12224444444444558888888810···102020202020···2040···40
size11112288101010104040222222101010102···244448···84···4

50 irreducible representations

dim11111122222222444
type++++++++++-++++-
imageC1C2C2C2C2C2D4D4D4D5Q16D10D10C5⋊D4D4×D5D4×D5D5×Q16
kernelC40.26D4C8×Dic5C2×Dic20C2×C5⋊Q16Dic5⋊Q8C10×Q16C52C8C40C2×Dic5C2×Q16Dic5C2×C8C2×Q8C8C4C22C2
# reps11122122228248228

Matrix representation of C40.26D4 in GL4(𝔽41) generated by

403400
7700
00022
001324
,
9000
193200
004036
00251
,
303200
271100
002927
002512
G:=sub<GL(4,GF(41))| [40,7,0,0,34,7,0,0,0,0,0,13,0,0,22,24],[9,19,0,0,0,32,0,0,0,0,40,25,0,0,36,1],[30,27,0,0,32,11,0,0,0,0,29,25,0,0,27,12] >;

C40.26D4 in GAP, Magma, Sage, TeX

C_{40}._{26}D_4
% in TeX

G:=Group("C40.26D4");
// GroupNames label

G:=SmallGroup(320,808);
// by ID

G=gap.SmallGroup(320,808);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,232,422,135,184,570,297,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^4=1,c^2=a^20,b*a*b^-1=a^9,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations

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