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## G = C2×C20.8Q8order 320 = 26·5

### Direct product of C2 and C20.8Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C20.8Q8
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4×Dic5 — C2×C4×Dic5 — C2×C20.8Q8
 Lower central C5 — C10 — C2×C20.8Q8
 Upper central C1 — C22×C4 — C22×C8

Generators and relations for C2×C20.8Q8
G = < a,b,c,d | a2=b20=1, c4=b10, d2=b15c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b9, dcd-1=b15c3 >

Subgroups: 334 in 138 conjugacy classes, 87 normal (29 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C42, C2×C8, C2×C8, C22×C4, C22×C4, Dic5, Dic5, C20, C20, C2×C10, C2×C10, C4⋊C8, C2×C42, C22×C8, C22×C8, C52C8, C40, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C2×C4⋊C8, C2×C52C8, C2×C52C8, C4×Dic5, C2×C40, C2×C40, C22×Dic5, C22×C20, C20.8Q8, C22×C52C8, C2×C4×Dic5, C22×C40, C2×C20.8Q8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, D5, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, D10, C4⋊C8, C2×C4⋊C4, C22×C8, C2×M4(2), Dic10, C4×D5, C5⋊D4, C22×D5, C2×C4⋊C8, C8×D5, C8⋊D5, C10.D4, C2×Dic10, C2×C4×D5, C2×C5⋊D4, C20.8Q8, D5×C2×C8, C2×C8⋊D5, C2×C10.D4, C2×C20.8Q8

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

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

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

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

104 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4P 5A 5B 8A ··· 8H 8I ··· 8P 10A ··· 10N 20A ··· 20P 40A ··· 40AF order 1 2 ··· 2 4 ··· 4 4 ··· 4 5 5 8 ··· 8 8 ··· 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 ··· 1 1 ··· 1 10 ··· 10 2 2 2 ··· 2 10 ··· 10 2 ··· 2 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + + + - image C1 C2 C2 C2 C2 C4 C4 C8 D4 Q8 D5 M4(2) D10 D10 Dic10 C4×D5 C5⋊D4 C4×D5 C8×D5 C8⋊D5 kernel C2×C20.8Q8 C20.8Q8 C22×C5⋊2C8 C2×C4×Dic5 C22×C40 C4×Dic5 C22×Dic5 C2×Dic5 C2×C20 C2×C20 C22×C8 C2×C10 C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 1 1 1 4 4 16 2 2 2 4 4 2 8 4 8 4 16 16

Matrix representation of C2×C20.8Q8 in GL5(𝔽41)

 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1
,
 40 0 0 0 0 0 0 9 0 0 0 32 13 0 0 0 0 0 0 9 0 0 0 32 19
,
 9 0 0 0 0 0 38 0 0 0 0 0 38 0 0 0 0 0 33 27 0 0 0 14 8
,
 1 0 0 0 0 0 13 2 0 0 0 39 28 0 0 0 0 0 29 14 0 0 0 16 12

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,0,32,0,0,0,9,13,0,0,0,0,0,0,32,0,0,0,9,19],[9,0,0,0,0,0,38,0,0,0,0,0,38,0,0,0,0,0,33,14,0,0,0,27,8],[1,0,0,0,0,0,13,39,0,0,0,2,28,0,0,0,0,0,29,16,0,0,0,14,12] >;

C2×C20.8Q8 in GAP, Magma, Sage, TeX

C_2\times C_{20}._8Q_8
% in TeX

G:=Group("C2xC20.8Q8");
// GroupNames label

G:=SmallGroup(320,726);
// by ID

G=gap.SmallGroup(320,726);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,422,58,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=1,c^4=b^10,d^2=b^15*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^9,d*c*d^-1=b^15*c^3>;
// generators/relations

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