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G = C2×C20⋊Q8order 320 = 26·5

Direct product of C2 and C20⋊Q8

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C20⋊Q8, C204(C2×Q8), (C2×C20)⋊2Q8, C102(C4⋊Q8), (C2×Dic5)⋊8Q8, Dic51(C2×Q8), C42(C2×Dic10), (C2×C4)⋊6Dic10, C4⋊C4.258D10, C22.30(Q8×D5), C10.7(C22×Q8), (C2×C10).41C24, Dic5.14(C2×D4), C22.129(D4×D5), C10.39(C22×D4), (C2×C20).132C23, (C2×Dic5).165D4, (C22×C4).173D10, C2.9(C22×Dic10), C22.79(C23×D5), C4⋊Dic5.356C22, (C2×Dic5).13C23, (C22×Dic10).9C2, C22.36(C2×Dic10), C23.321(C22×D5), (C22×C10).390C23, (C22×C20).214C22, (C4×Dic5).278C22, (C2×Dic10).234C22, C10.D4.102C22, (C22×Dic5).229C22, C52(C2×C4⋊Q8), C2.5(C2×Q8×D5), C2.13(C2×D4×D5), (C2×C4⋊C4).24D5, (C10×C4⋊C4).17C2, (C2×C10).50(C2×Q8), (C2×C4×Dic5).12C2, (C2×C10).385(C2×D4), (C2×C4⋊Dic5).43C2, (C5×C4⋊C4).290C22, (C2×C4).137(C22×D5), (C2×C10.D4).23C2, SmallGroup(320,1169)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C2×C20⋊Q8
C1C5C10C2×C10C2×Dic5C22×Dic5C2×C4×Dic5 — C2×C20⋊Q8
C5C2×C10 — C2×C20⋊Q8
C1C23C2×C4⋊C4

Generators and relations for C2×C20⋊Q8
 G = < a,b,c,d | a2=b20=c4=1, d2=c2, ab=ba, ac=ca, ad=da, cbc-1=b11, dbd-1=b9, dcd-1=c-1 >

Subgroups: 894 in 290 conjugacy classes, 143 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×16], C22, C22 [×6], C5, C2×C4 [×10], C2×C4 [×24], Q8 [×16], C23, C10 [×3], C10 [×4], C42 [×4], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×16], Dic5 [×8], Dic5 [×4], C20 [×4], C20 [×4], C2×C10, C2×C10 [×6], C2×C42, C2×C4⋊C4, C2×C4⋊C4 [×3], C4⋊Q8 [×8], C22×Q8 [×2], Dic10 [×16], C2×Dic5 [×16], C2×Dic5 [×4], C2×C20 [×10], C2×C20 [×4], C22×C10, C2×C4⋊Q8, C4×Dic5 [×4], C10.D4 [×8], C4⋊Dic5 [×4], C5×C4⋊C4 [×4], C2×Dic10 [×8], C2×Dic10 [×8], C22×Dic5 [×2], C22×Dic5 [×2], C22×C20, C22×C20 [×2], C20⋊Q8 [×8], C2×C4×Dic5, C2×C10.D4 [×2], C2×C4⋊Dic5, C10×C4⋊C4, C22×Dic10 [×2], C2×C20⋊Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], D5, C2×D4 [×6], C2×Q8 [×12], C24, D10 [×7], C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], Dic10 [×4], C22×D5 [×7], C2×C4⋊Q8, C2×Dic10 [×6], D4×D5 [×2], Q8×D5 [×2], C23×D5, C20⋊Q8 [×4], C22×Dic10, C2×D4×D5, C2×Q8×D5, C2×C20⋊Q8

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

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

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

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

68 conjugacy classes

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

68 irreducible representations

dim1111111222222244
type++++++++--+++-+-
imageC1C2C2C2C2C2C2D4Q8Q8D5D10D10Dic10D4×D5Q8×D5
kernelC2×C20⋊Q8C20⋊Q8C2×C4×Dic5C2×C10.D4C2×C4⋊Dic5C10×C4⋊C4C22×Dic10C2×Dic5C2×Dic5C2×C20C2×C4⋊C4C4⋊C4C22×C4C2×C4C22C22
# reps18121124442861644

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

100000
010000
0040000
0004000
000010
000001
,
4010000
3370000
0004000
0013500
00003935
0000352
,
11280000
22300000
001000
000100
0000352
000026
,
38210000
2130000
00283900
0021300
000026
0000639

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],[40,33,0,0,0,0,1,7,0,0,0,0,0,0,0,1,0,0,0,0,40,35,0,0,0,0,0,0,39,35,0,0,0,0,35,2],[11,22,0,0,0,0,28,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,2,0,0,0,0,2,6],[38,21,0,0,0,0,21,3,0,0,0,0,0,0,28,2,0,0,0,0,39,13,0,0,0,0,0,0,2,6,0,0,0,0,6,39] >;

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(320,1169);
// by ID

G=gap.SmallGroup(320,1169);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,184,675,297,80,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^11,d*b*d^-1=b^9,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽