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## G = C2×C20⋊2Q8order 320 = 26·5

### Direct product of C2 and C20⋊2Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×C20⋊2Q8
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C22×Dic10 — C2×C20⋊2Q8
 Lower central C5 — C2×C10 — C2×C20⋊2Q8
 Upper central C1 — C23 — C2×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, C4, C4, C22, C22, C5, C2×C4, C2×C4, Q8, C23, C10, C10, C42, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C2×C42, C2×C4⋊C4, C4⋊Q8, C22×Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C22×C10, C2×C4⋊Q8, C4⋊Dic5, C4×C20, C2×Dic10, C2×Dic10, C22×Dic5, C22×C20, C22×C20, C202Q8, C2×C4⋊Dic5, C2×C4×C20, C22×Dic10, C2×C202Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C24, D10, C4⋊Q8, C22×D4, C22×Q8, Dic10, D20, C22×D5, C2×C4⋊Q8, C2×Dic10, C2×D20, C23×D5, C202Q8, C22×Dic10, C22×D20, C2×C202Q8

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

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

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

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

92 conjugacy classes

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

92 irreducible representations

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

Matrix representation of C2×C202Q8 in GL5(𝔽41)

 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 1 0 0 0 40 0 0 0 0 0 0 1 40 0 0 0 36 6
,
 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 39 32 0 0 0 37 2
,
 1 0 0 0 0 0 25 27 0 0 0 27 16 0 0 0 0 0 27 27 0 0 0 17 14

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

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