Copied to
clipboard

## G = C2×C20.Q8order 320 = 26·5

### Direct product of C2 and C20.Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×C20.Q8
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C5⋊2C8 — C22×C5⋊2C8 — C2×C20.Q8
 Lower central C5 — C10 — C20 — C2×C20.Q8
 Upper central C1 — C23 — C22×C4 — C2×C4⋊C4

Generators and relations for C2×C20.Q8
G = < a,b,c,d | a2=b20=c4=1, d2=b5c2, ab=ba, ac=ca, ad=da, cbc-1=b11, dbd-1=b9, dcd-1=b15c-1 >

Subgroups: 366 in 130 conjugacy classes, 79 normal (27 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, Dic5, C20, C20, C20, C2×C10, C2×C10, C4.Q8, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C52C8, C2×Dic5, C2×C20, C2×C20, C2×C20, C22×C10, C2×C4.Q8, C2×C52C8, C4⋊Dic5, C4⋊Dic5, C5×C4⋊C4, C5×C4⋊C4, C22×Dic5, C22×C20, C22×C20, C20.Q8, C22×C52C8, C2×C4⋊Dic5, C10×C4⋊C4, C2×C20.Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, D10, C4.Q8, C2×C4⋊C4, C2×SD16, Dic10, C4×D5, C5⋊D4, C22×D5, C2×C4.Q8, C10.D4, D4.D5, Q8⋊D5, C2×Dic10, C2×C4×D5, C2×C5⋊D4, C20.Q8, C2×C10.D4, C2×D4.D5, C2×Q8⋊D5, C2×C20.Q8

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

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 8A ··· 8H 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 4 4 4 4 4 4 4 4 4 4 4 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 2 2 2 4 4 4 4 20 20 20 20 2 2 10 ··· 10 2 ··· 2 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + - + + + + - - + image C1 C2 C2 C2 C2 C4 D4 Q8 D4 D5 SD16 D10 D10 Dic10 C4×D5 C5⋊D4 C5⋊D4 D4.D5 Q8⋊D5 kernel C2×C20.Q8 C20.Q8 C22×C5⋊2C8 C2×C4⋊Dic5 C10×C4⋊C4 C2×C5⋊2C8 C2×C20 C2×C20 C22×C10 C2×C4⋊C4 C2×C10 C4⋊C4 C22×C4 C2×C4 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 1 1 1 8 1 2 1 2 8 4 2 8 8 4 4 4 4

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

 40 0 0 0 0 0 0 40 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
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 1 2 0 0 0 0 40 40 0 0 0 0 0 0 1 40 0 0 0 0 36 6
,
 25 14 0 0 0 0 14 16 0 0 0 0 0 0 11 13 0 0 0 0 16 30 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 15 26 0 0 0 0 15 15 0 0 0 0 0 0 0 11 0 0 0 0 15 30 0 0 0 0 0 0 6 1 0 0 0 0 6 35

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,40,0,0,0,0,2,40,0,0,0,0,0,0,1,36,0,0,0,0,40,6],[25,14,0,0,0,0,14,16,0,0,0,0,0,0,11,16,0,0,0,0,13,30,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[15,15,0,0,0,0,26,15,0,0,0,0,0,0,0,15,0,0,0,0,11,30,0,0,0,0,0,0,6,6,0,0,0,0,1,35] >;

C2×C20.Q8 in GAP, Magma, Sage, TeX

C_2\times C_{20}.Q_8
% in TeX

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

G:=SmallGroup(320,590);
// by ID

G=gap.SmallGroup(320,590);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,1094,58,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^4=1,d^2=b^5*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=b^15*c^-1>;
// generators/relations

׿
×
𝔽