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## G = C2×C20.6Q8order 320 = 26·5

### Direct product of C2 and C20.6Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×C20.6Q8
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C2×C10.D4 — C2×C20.6Q8
 Lower central C5 — C2×C10 — C2×C20.6Q8
 Upper central C1 — C23 — C2×C42

Generators and relations for C2×C20.6Q8
G = < a,b,c,d | a2=b20=c4=1, d2=b10c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b10c-1 >

Subgroups: 606 in 226 conjugacy classes, 127 normal (13 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×12], C22, C22 [×6], C5, C2×C4 [×10], C2×C4 [×20], C23, C10, C10 [×6], C42 [×4], C4⋊C4 [×24], C22×C4, C22×C4 [×2], C22×C4 [×4], Dic5 [×8], C20 [×4], C20 [×4], C2×C10, C2×C10 [×6], C2×C42, C2×C4⋊C4 [×6], C42.C2 [×8], C2×Dic5 [×8], C2×Dic5 [×8], C2×C20 [×10], C2×C20 [×4], C22×C10, C2×C42.C2, C10.D4 [×16], C4⋊Dic5 [×8], C4×C20 [×4], C22×Dic5 [×4], C22×C20, C22×C20 [×2], C20.6Q8 [×8], C2×C10.D4 [×4], C2×C4⋊Dic5 [×2], C2×C4×C20, C2×C20.6Q8
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×4], C24, D10 [×7], C42.C2 [×4], C22×Q8, C2×C4○D4 [×2], Dic10 [×4], C22×D5 [×7], C2×C42.C2, C2×Dic10 [×6], C4○D20 [×4], C23×D5, C20.6Q8 [×4], C22×Dic10, C2×C4○D20 [×2], C2×C20.6Q8

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

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

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

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

92 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4L 4M ··· 4T 5A 5B 10A ··· 10N 20A ··· 20AV order 1 2 ··· 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 ··· 2 20 ··· 20 2 2 2 ··· 2 2 ··· 2

92 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + - + + + - image C1 C2 C2 C2 C2 Q8 D5 C4○D4 D10 D10 Dic10 C4○D20 kernel C2×C20.6Q8 C20.6Q8 C2×C10.D4 C2×C4⋊Dic5 C2×C4×C20 C2×C20 C2×C42 C2×C10 C42 C22×C4 C2×C4 C22 # reps 1 8 4 2 1 4 2 8 8 6 16 32

Matrix representation of C2×C20.6Q8 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 2 30 0 0 0 0 27 16 0 0 0 0 0 0 32 30 0 0 0 0 11 27 0 0 0 0 0 0 9 11 0 0 0 0 30 14
,
 18 40 0 0 0 0 36 23 0 0 0 0 0 0 24 1 0 0 0 0 40 17 0 0 0 0 0 0 32 0 0 0 0 0 0 32
,
 39 25 0 0 0 0 8 2 0 0 0 0 0 0 16 12 0 0 0 0 23 25 0 0 0 0 0 0 26 38 0 0 0 0 20 15

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,27,0,0,0,0,30,16,0,0,0,0,0,0,32,11,0,0,0,0,30,27,0,0,0,0,0,0,9,30,0,0,0,0,11,14],[18,36,0,0,0,0,40,23,0,0,0,0,0,0,24,40,0,0,0,0,1,17,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[39,8,0,0,0,0,25,2,0,0,0,0,0,0,16,23,0,0,0,0,12,25,0,0,0,0,0,0,26,20,0,0,0,0,38,15] >;

C2×C20.6Q8 in GAP, Magma, Sage, TeX

C_2\times C_{20}._6Q_8
% in TeX

G:=Group("C2xC20.6Q8");
// GroupNames label

G:=SmallGroup(320,1141);
// by ID

G=gap.SmallGroup(320,1141);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,100,675,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^4=1,d^2=b^10*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^10*c^-1>;
// generators/relations

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