Copied to
clipboard

G = C2×C202Q8order 320 = 26·5

Direct product of C2 and C202Q8

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

Aliases: C2×C202Q8, C42.272D10, C205(C2×Q8), C101(C4⋊Q8), (C2×C20)⋊12Q8, C43(C2×Dic10), (C2×C4)⋊9Dic10, C4.42(C2×D20), (C2×C4).97D20, C20.285(C2×D4), (C2×C20).388D4, C10.1(C22×D4), (C2×C42).19D5, C2.4(C22×D20), C10.2(C22×Q8), (C2×C10).12C24, C22.62(C2×D20), (C4×C20).312C22, (C2×C20).778C23, (C22×C4).434D10, (C2×Dic5).1C23, C2.4(C22×Dic10), C22.59(C23×D5), C4⋊Dic5.286C22, (C22×Dic10).8C2, C22.34(C2×Dic10), C23.310(C22×D5), (C22×C10).374C23, (C22×C20).521C22, (C2×Dic10).230C22, (C22×Dic5).72C22, C51(C2×C4⋊Q8), (C2×C4×C20).13C2, (C2×C10).46(C2×Q8), (C2×C10).168(C2×D4), (C2×C4⋊Dic5).25C2, (C2×C4).727(C22×D5), SmallGroup(320,1140)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C2×C202Q8
C1C5C10C2×C10C2×Dic5C22×Dic5C22×Dic10 — C2×C202Q8
C5C2×C10 — C2×C202Q8
C1C23C2×C42

Generators and relations for C2×C202Q8
 G = < a,b,c,d | a2=b20=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 894 in 290 conjugacy classes, 159 normal (13 characteristic)
C1, C2, C2 [×6], C4 [×12], C4 [×8], C22, C22 [×6], C5, C2×C4 [×18], C2×C4 [×16], Q8 [×16], C23, C10, C10 [×6], C42 [×4], C4⋊C4 [×16], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×16], Dic5 [×8], C20 [×12], C2×C10, C2×C10 [×6], C2×C42, C2×C4⋊C4 [×4], C4⋊Q8 [×8], C22×Q8 [×2], Dic10 [×16], C2×Dic5 [×8], C2×Dic5 [×8], C2×C20 [×18], C22×C10, C2×C4⋊Q8, C4⋊Dic5 [×16], C4×C20 [×4], C2×Dic10 [×8], C2×Dic10 [×8], C22×Dic5 [×4], C22×C20, C22×C20 [×2], C202Q8 [×8], C2×C4⋊Dic5 [×4], C2×C4×C20, C22×Dic10 [×2], C2×C202Q8
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 [×8], D20 [×4], C22×D5 [×7], C2×C4⋊Q8, C2×Dic10 [×12], C2×D20 [×6], C23×D5, C202Q8 [×4], C22×Dic10 [×2], C22×D20, C2×C202Q8

Smallest permutation representation of C2×C202Q8
Regular action on 320 points
Generators in S320
(1 183)(2 184)(3 185)(4 186)(5 187)(6 188)(7 189)(8 190)(9 191)(10 192)(11 193)(12 194)(13 195)(14 196)(15 197)(16 198)(17 199)(18 200)(19 181)(20 182)(21 211)(22 212)(23 213)(24 214)(25 215)(26 216)(27 217)(28 218)(29 219)(30 220)(31 201)(32 202)(33 203)(34 204)(35 205)(36 206)(37 207)(38 208)(39 209)(40 210)(41 245)(42 246)(43 247)(44 248)(45 249)(46 250)(47 251)(48 252)(49 253)(50 254)(51 255)(52 256)(53 257)(54 258)(55 259)(56 260)(57 241)(58 242)(59 243)(60 244)(61 285)(62 286)(63 287)(64 288)(65 289)(66 290)(67 291)(68 292)(69 293)(70 294)(71 295)(72 296)(73 297)(74 298)(75 299)(76 300)(77 281)(78 282)(79 283)(80 284)(81 314)(82 315)(83 316)(84 317)(85 318)(86 319)(87 320)(88 301)(89 302)(90 303)(91 304)(92 305)(93 306)(94 307)(95 308)(96 309)(97 310)(98 311)(99 312)(100 313)(101 221)(102 222)(103 223)(104 224)(105 225)(106 226)(107 227)(108 228)(109 229)(110 230)(111 231)(112 232)(113 233)(114 234)(115 235)(116 236)(117 237)(118 238)(119 239)(120 240)(121 146)(122 147)(123 148)(124 149)(125 150)(126 151)(127 152)(128 153)(129 154)(130 155)(131 156)(132 157)(133 158)(134 159)(135 160)(136 141)(137 142)(138 143)(139 144)(140 145)(161 276)(162 277)(163 278)(164 279)(165 280)(166 261)(167 262)(168 263)(169 264)(170 265)(171 266)(172 267)(173 268)(174 269)(175 270)(176 271)(177 272)(178 273)(179 274)(180 275)
(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)
(1 263 206 255)(2 264 207 256)(3 265 208 257)(4 266 209 258)(5 267 210 259)(6 268 211 260)(7 269 212 241)(8 270 213 242)(9 271 214 243)(10 272 215 244)(11 273 216 245)(12 274 217 246)(13 275 218 247)(14 276 219 248)(15 277 220 249)(16 278 201 250)(17 279 202 251)(18 280 203 252)(19 261 204 253)(20 262 205 254)(21 56 188 173)(22 57 189 174)(23 58 190 175)(24 59 191 176)(25 60 192 177)(26 41 193 178)(27 42 194 179)(28 43 195 180)(29 44 196 161)(30 45 197 162)(31 46 198 163)(32 47 199 164)(33 48 200 165)(34 49 181 166)(35 50 182 167)(36 51 183 168)(37 52 184 169)(38 53 185 170)(39 54 186 171)(40 55 187 172)(61 82 143 118)(62 83 144 119)(63 84 145 120)(64 85 146 101)(65 86 147 102)(66 87 148 103)(67 88 149 104)(68 89 150 105)(69 90 151 106)(70 91 152 107)(71 92 153 108)(72 93 154 109)(73 94 155 110)(74 95 156 111)(75 96 157 112)(76 97 158 113)(77 98 159 114)(78 99 160 115)(79 100 141 116)(80 81 142 117)(121 221 288 318)(122 222 289 319)(123 223 290 320)(124 224 291 301)(125 225 292 302)(126 226 293 303)(127 227 294 304)(128 228 295 305)(129 229 296 306)(130 230 297 307)(131 231 298 308)(132 232 299 309)(133 233 300 310)(134 234 281 311)(135 235 282 312)(136 236 283 313)(137 237 284 314)(138 238 285 315)(139 239 286 316)(140 240 287 317)
(1 229 206 306)(2 228 207 305)(3 227 208 304)(4 226 209 303)(5 225 210 302)(6 224 211 301)(7 223 212 320)(8 222 213 319)(9 221 214 318)(10 240 215 317)(11 239 216 316)(12 238 217 315)(13 237 218 314)(14 236 219 313)(15 235 220 312)(16 234 201 311)(17 233 202 310)(18 232 203 309)(19 231 204 308)(20 230 205 307)(21 88 188 104)(22 87 189 103)(23 86 190 102)(24 85 191 101)(25 84 192 120)(26 83 193 119)(27 82 194 118)(28 81 195 117)(29 100 196 116)(30 99 197 115)(31 98 198 114)(32 97 199 113)(33 96 200 112)(34 95 181 111)(35 94 182 110)(36 93 183 109)(37 92 184 108)(38 91 185 107)(39 90 186 106)(40 89 187 105)(41 62 178 144)(42 61 179 143)(43 80 180 142)(44 79 161 141)(45 78 162 160)(46 77 163 159)(47 76 164 158)(48 75 165 157)(49 74 166 156)(50 73 167 155)(51 72 168 154)(52 71 169 153)(53 70 170 152)(54 69 171 151)(55 68 172 150)(56 67 173 149)(57 66 174 148)(58 65 175 147)(59 64 176 146)(60 63 177 145)(121 243 288 271)(122 242 289 270)(123 241 290 269)(124 260 291 268)(125 259 292 267)(126 258 293 266)(127 257 294 265)(128 256 295 264)(129 255 296 263)(130 254 297 262)(131 253 298 261)(132 252 299 280)(133 251 300 279)(134 250 281 278)(135 249 282 277)(136 248 283 276)(137 247 284 275)(138 246 285 274)(139 245 286 273)(140 244 287 272)

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

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

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

92 conjugacy classes

class 1 2A···2G4A···4L4M···4T5A5B10A···10N20A···20AV
order12···24···44···45510···1020···20
size11···12···220···20222···22···2

92 irreducible representations

dim111112222222
type++++++-+++-+
imageC1C2C2C2C2D4Q8D5D10D10Dic10D20
kernelC2×C202Q8C202Q8C2×C4⋊Dic5C2×C4×C20C22×Dic10C2×C20C2×C20C2×C42C42C22×C4C2×C4C2×C4
# reps18412482863216

Matrix representation of C2×C202Q8 in GL5(𝔽41)

400000
01000
00100
000400
000040
,
400000
00100
040000
000140
000366
,
400000
01000
00100
0003932
000372
,
10000
0252700
0271600
0002727
0001714

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,1,36,0,0,0,40,6],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,39,37,0,0,0,32,2],[1,0,0,0,0,0,25,27,0,0,0,27,16,0,0,0,0,0,27,17,0,0,0,27,14] >;

C2×C202Q8 in GAP, Magma, Sage, TeX

C_2\times C_{20}\rtimes_2Q_8
% in TeX

G:=Group("C2xC20:2Q8");
// GroupNames label

G:=SmallGroup(320,1140);
// by ID

G=gap.SmallGroup(320,1140);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,184,675,80,12550]);
// 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,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽