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G = (C2×C20)⋊Q8order 320 = 26·5

6th semidirect product of C2×C20 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C20)⋊6Q8, (C2×Dic5)⋊5Q8, (C2×C4)⋊5Dic10, C2.1(C20⋊Q8), C10.9(C4⋊Q8), C10.10(C4×Q8), C22.9(Q8×D5), C2.4(C4×Dic10), C22.50(D4×D5), (C22×C4).5D10, (C2×Dic10)⋊15C4, C10.1(C22⋊Q8), C2.1(D10⋊Q8), (C2×Dic5).181D4, C2.C42.7D5, C2.4(Dic53Q8), C10.14(C4.4D4), C22.25(C4○D20), (C22×Dic10).1C2, C22.12(C2×Dic10), C23.243(C22×D5), C22.25(D42D5), Dic5.18(C22⋊C4), (C22×C20).326C22, (C22×C10).271C23, C52(C23.67C23), C2.1(Dic5.5D4), (C22×Dic5).1C22, C10.10C42.30C2, C2.1(Dic5.14D4), (C2×C4).22(C4×D5), C2.5(D5×C22⋊C4), C22.80(C2×C4×D5), (C2×C10).56(C2×Q8), (C2×C4×Dic5).20C2, (C2×C20).203(C2×C4), (C2×C10).187(C2×D4), C10.42(C2×C22⋊C4), (C2×Dic5).84(C2×C4), (C2×C10.D4).3C2, (C2×C10).120(C4○D4), (C2×C10).138(C22×C4), (C5×C2.C42).14C2, SmallGroup(320,273)

Series: Derived Chief Lower central Upper central

C1C2×C10 — (C2×C20)⋊Q8
C1C5C10C2×C10C22×C10C22×Dic5C22×Dic10 — (C2×C20)⋊Q8
C5C2×C10 — (C2×C20)⋊Q8
C1C23C2.C42

Generators and relations for (C2×C20)⋊Q8
 G = < a,b,c,d | a2=b20=c4=1, d2=c2, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=b9, dcd-1=c-1 >

Subgroups: 622 in 186 conjugacy classes, 77 normal (51 characteristic)
C1, C2 [×7], C4 [×14], C22 [×7], C5, C2×C4 [×4], C2×C4 [×24], Q8 [×8], C23, C10 [×7], C42 [×2], C4⋊C4 [×2], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×8], Dic5 [×4], Dic5 [×5], C20 [×5], C2×C10 [×7], C2.C42, C2.C42 [×3], C2×C42, C2×C4⋊C4, C22×Q8, Dic10 [×8], C2×Dic5 [×10], C2×Dic5 [×7], C2×C20 [×4], C2×C20 [×7], C22×C10, C23.67C23, C4×Dic5 [×2], C10.D4 [×2], C2×Dic10 [×4], C2×Dic10 [×4], C22×Dic5 [×4], C22×C20 [×3], C10.10C42 [×3], C5×C2.C42, C2×C4×Dic5, C2×C10.D4, C22×Dic10, (C2×C20)⋊Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], D10 [×3], C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4, C4⋊Q8, Dic10 [×2], C4×D5 [×2], C22×D5, C23.67C23, C2×Dic10, C2×C4×D5, C4○D20, D4×D5 [×2], D42D5, Q8×D5, C4×Dic10, Dic5.14D4, D5×C22⋊C4, Dic5.5D4, Dic53Q8, C20⋊Q8, D10⋊Q8, (C2×C20)⋊Q8

Smallest permutation representation of (C2×C20)⋊Q8
Regular action on 320 points
Generators in S320
(1 149)(2 150)(3 151)(4 152)(5 153)(6 154)(7 155)(8 156)(9 157)(10 158)(11 159)(12 160)(13 141)(14 142)(15 143)(16 144)(17 145)(18 146)(19 147)(20 148)(21 125)(22 126)(23 127)(24 128)(25 129)(26 130)(27 131)(28 132)(29 133)(30 134)(31 135)(32 136)(33 137)(34 138)(35 139)(36 140)(37 121)(38 122)(39 123)(40 124)(41 116)(42 117)(43 118)(44 119)(45 120)(46 101)(47 102)(48 103)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(57 112)(58 113)(59 114)(60 115)(61 284)(62 285)(63 286)(64 287)(65 288)(66 289)(67 290)(68 291)(69 292)(70 293)(71 294)(72 295)(73 296)(74 297)(75 298)(76 299)(77 300)(78 281)(79 282)(80 283)(81 237)(82 238)(83 239)(84 240)(85 221)(86 222)(87 223)(88 224)(89 225)(90 226)(91 227)(92 228)(93 229)(94 230)(95 231)(96 232)(97 233)(98 234)(99 235)(100 236)(161 208)(162 209)(163 210)(164 211)(165 212)(166 213)(167 214)(168 215)(169 216)(170 217)(171 218)(172 219)(173 220)(174 201)(175 202)(176 203)(177 204)(178 205)(179 206)(180 207)(181 268)(182 269)(183 270)(184 271)(185 272)(186 273)(187 274)(188 275)(189 276)(190 277)(191 278)(192 279)(193 280)(194 261)(195 262)(196 263)(197 264)(198 265)(199 266)(200 267)(241 319)(242 320)(243 301)(244 302)(245 303)(246 304)(247 305)(248 306)(249 307)(250 308)(251 309)(252 310)(253 311)(254 312)(255 313)(256 314)(257 315)(258 316)(259 317)(260 318)
(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)
(1 192 50 220)(2 280 51 174)(3 194 52 202)(4 262 53 176)(5 196 54 204)(6 264 55 178)(7 198 56 206)(8 266 57 180)(9 200 58 208)(10 268 59 162)(11 182 60 210)(12 270 41 164)(13 184 42 212)(14 272 43 166)(15 186 44 214)(16 274 45 168)(17 188 46 216)(18 276 47 170)(19 190 48 218)(20 278 49 172)(21 221 76 248)(22 86 77 307)(23 223 78 250)(24 88 79 309)(25 225 80 252)(26 90 61 311)(27 227 62 254)(28 92 63 313)(29 229 64 256)(30 94 65 315)(31 231 66 258)(32 96 67 317)(33 233 68 260)(34 98 69 319)(35 235 70 242)(36 100 71 301)(37 237 72 244)(38 82 73 303)(39 239 74 246)(40 84 75 305)(81 295 302 121)(83 297 304 123)(85 299 306 125)(87 281 308 127)(89 283 310 129)(91 285 312 131)(93 287 314 133)(95 289 316 135)(97 291 318 137)(99 293 320 139)(101 169 145 275)(102 217 146 189)(103 171 147 277)(104 219 148 191)(105 173 149 279)(106 201 150 193)(107 175 151 261)(108 203 152 195)(109 177 153 263)(110 205 154 197)(111 179 155 265)(112 207 156 199)(113 161 157 267)(114 209 158 181)(115 163 159 269)(116 211 160 183)(117 165 141 271)(118 213 142 185)(119 167 143 273)(120 215 144 187)(122 238 296 245)(124 240 298 247)(126 222 300 249)(128 224 282 251)(130 226 284 253)(132 228 286 255)(134 230 288 257)(136 232 290 259)(138 234 292 241)(140 236 294 243)
(1 317 50 96)(2 306 51 85)(3 315 52 94)(4 304 53 83)(5 313 54 92)(6 302 55 81)(7 311 56 90)(8 320 57 99)(9 309 58 88)(10 318 59 97)(11 307 60 86)(12 316 41 95)(13 305 42 84)(14 314 43 93)(15 303 44 82)(16 312 45 91)(17 301 46 100)(18 310 47 89)(19 319 48 98)(20 308 49 87)(21 193 76 201)(22 182 77 210)(23 191 78 219)(24 200 79 208)(25 189 80 217)(26 198 61 206)(27 187 62 215)(28 196 63 204)(29 185 64 213)(30 194 65 202)(31 183 66 211)(32 192 67 220)(33 181 68 209)(34 190 69 218)(35 199 70 207)(36 188 71 216)(37 197 72 205)(38 186 73 214)(39 195 74 203)(40 184 75 212)(101 236 145 243)(102 225 146 252)(103 234 147 241)(104 223 148 250)(105 232 149 259)(106 221 150 248)(107 230 151 257)(108 239 152 246)(109 228 153 255)(110 237 154 244)(111 226 155 253)(112 235 156 242)(113 224 157 251)(114 233 158 260)(115 222 159 249)(116 231 160 258)(117 240 141 247)(118 229 142 256)(119 238 143 245)(120 227 144 254)(121 264 295 178)(122 273 296 167)(123 262 297 176)(124 271 298 165)(125 280 299 174)(126 269 300 163)(127 278 281 172)(128 267 282 161)(129 276 283 170)(130 265 284 179)(131 274 285 168)(132 263 286 177)(133 272 287 166)(134 261 288 175)(135 270 289 164)(136 279 290 173)(137 268 291 162)(138 277 292 171)(139 266 293 180)(140 275 294 169)

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

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

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

68 conjugacy classes

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

68 irreducible representations

dim1111111222222222444
type+++++++--++-+--
imageC1C2C2C2C2C2C4D4Q8Q8D5C4○D4D10Dic10C4×D5C4○D20D4×D5D42D5Q8×D5
kernel(C2×C20)⋊Q8C10.10C42C5×C2.C42C2×C4×Dic5C2×C10.D4C22×Dic10C2×Dic10C2×Dic5C2×Dic5C2×C20C2.C42C2×C10C22×C4C2×C4C2×C4C22C22C22C22
# reps1311118422246888422

Matrix representation of (C2×C20)⋊Q8 in GL6(𝔽41)

100000
010000
001000
000100
0000400
0000040
,
100000
010000
00133200
009000
00004018
000091
,
2680000
23150000
001000
000100
00003612
0000395
,
2350000
35390000
0021300
00253900
0000400
0000040

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,9,0,0,0,0,32,0,0,0,0,0,0,0,40,9,0,0,0,0,18,1],[26,23,0,0,0,0,8,15,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,39,0,0,0,0,12,5],[2,35,0,0,0,0,35,39,0,0,0,0,0,0,2,25,0,0,0,0,13,39,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;

(C2×C20)⋊Q8 in GAP, Magma, Sage, TeX

(C_2\times C_{20})\rtimes Q_8
% in TeX

G:=Group("(C2xC20):Q8");
// GroupNames label

G:=SmallGroup(320,273);
// by ID

G=gap.SmallGroup(320,273);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,120,387,58,12550]);
// Polycyclic

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

׿
×
𝔽