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## G = C2.(C20⋊Q8)  order 320 = 26·5

### 2nd central stem extension by C2 of C20⋊Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C10 — C2.(C20⋊Q8)
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C2×C10.D4 — C2.(C20⋊Q8)
 Lower central C5 — C22×C10 — C2.(C20⋊Q8)
 Upper central C1 — C23 — C2.C42

Generators and relations for C2.(C20⋊Q8)
G = < a,b,c,d | a2=b20=c4=1, d2=c2, ab=ba, ac=ca, ad=da, cbc-1=ab11, dbd-1=ab9, dcd-1=ac-1 >

Subgroups: 502 in 150 conjugacy classes, 61 normal (51 characteristic)
C1, C2, C4, C22, C5, C2×C4, C2×C4, C23, C10, C4⋊C4, C22×C4, C22×C4, Dic5, C20, C2×C10, C2.C42, C2.C42, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.81C23, C10.D4, C4⋊Dic5, C22×Dic5, C22×C20, C10.10C42, C5×C2.C42, C2×C10.D4, C2×C4⋊Dic5, C2.(C20⋊Q8)
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C4⋊Q8, Dic10, C22×D5, C23.81C23, C2×Dic10, C4○D20, D4×D5, D42D5, Q8×D5, C20.6Q8, Dic5.14D4, D10.12D4, D10⋊D4, C20⋊Q8, Dic5.Q8, D10⋊Q8, C2.(C20⋊Q8)

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

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

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

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

62 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4F 4G ··· 4N 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 4 ··· 4 20 ··· 20 2 2 2 ··· 2 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + - - + + - + - - image C1 C2 C2 C2 C2 D4 Q8 Q8 D5 C4○D4 D10 Dic10 C4○D20 D4×D5 D4⋊2D5 Q8×D5 kernel C2.(C20⋊Q8) C10.10C42 C5×C2.C42 C2×C10.D4 C2×C4⋊Dic5 C2×Dic5 C2×Dic5 C2×C20 C2.C42 C2×C10 C22×C4 C2×C4 C22 C22 C22 C22 # reps 1 2 1 3 1 4 2 2 2 6 6 8 16 4 2 2

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

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 0 9 0 0 0 0 9 0 0 0 0 0 0 0 39 11 0 0 0 0 14 25 0 0 0 0 0 0 23 6 0 0 0 0 35 18
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 12 15 0 0 0 0 26 29 0 0 0 0 0 0 10 13 0 0 0 0 8 31 0 0 0 0 0 0 13 2 0 0 0 0 39 28

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,39,14,0,0,0,0,11,25,0,0,0,0,0,0,23,35,0,0,0,0,6,18],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,26,0,0,0,0,15,29,0,0,0,0,0,0,10,8,0,0,0,0,13,31,0,0,0,0,0,0,13,39,0,0,0,0,2,28] >;`

C2.(C20⋊Q8) in GAP, Magma, Sage, TeX

`C_2.(C_{20}\rtimes Q_8)`
`% in TeX`

`G:=Group("C2.(C20:Q8)");`
`// GroupNames label`

`G:=SmallGroup(320,284);`
`// by ID`

`G=gap.SmallGroup(320,284);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,64,254,387,100,12550]);`
`// Polycyclic`

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

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