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G = C2.(C20⋊Q8)  order 320 = 26·5

2nd central stem extension by C2 of C20⋊Q8

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C20).1Q8, C2.7(C20⋊Q8), C10.12(C4⋊Q8), (C2×Dic5).1Q8, (C2×C4).8Dic10, C22.39(Q8×D5), C10.2(C4⋊D4), (C2×Dic5).16D4, C22.151(D4×D5), (C22×C4).68D10, C2.8(D10⋊D4), C2.8(D10⋊Q8), C10.21(C22⋊Q8), (C22×C20).7C22, C2.4(C20.6Q8), C22.84(C4○D20), C10.11(C42.C2), C22.41(C2×Dic10), C2.C42.13D5, C23.353(C22×D5), C2.8(D10.12D4), C22.82(D42D5), C10.10C42.5C2, (C22×C10).282C23, C51(C23.81C23), C2.8(Dic5.Q8), C10.4(C22.D4), (C22×Dic5).7C22, C2.9(Dic5.14D4), (C2×C4⋊Dic5).8C2, (C2×C10).63(C2×Q8), (C2×C10).194(C2×D4), (C2×C10).57(C4○D4), (C2×C10.D4).8C2, (C5×C2.C42).9C2, SmallGroup(320,284)

Series: Derived Chief Lower central Upper central

C1C22×C10 — C2.(C20⋊Q8)
C1C5C10C2×C10C22×C10C22×Dic5C2×C10.D4 — C2.(C20⋊Q8)
C5C22×C10 — C2.(C20⋊Q8)
C1C23C2.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 [×7], C4 [×11], C22 [×7], C5, C2×C4 [×2], C2×C4 [×23], C23, C10 [×7], C4⋊C4 [×8], C22×C4 [×3], C22×C4 [×4], Dic5 [×7], C20 [×4], C2×C10 [×7], C2.C42, C2.C42 [×2], C2×C4⋊C4 [×4], C2×Dic5 [×6], C2×Dic5 [×9], C2×C20 [×2], C2×C20 [×8], C22×C10, C23.81C23, C10.D4 [×6], C4⋊Dic5 [×2], C22×Dic5 [×4], C22×C20 [×3], C10.10C42 [×2], C5×C2.C42, C2×C10.D4 [×3], C2×C4⋊Dic5, C2.(C20⋊Q8)
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], Q8 [×4], C23, D5, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×3], D10 [×3], C4⋊D4, C22⋊Q8 [×2], C22.D4, C42.C2 [×2], C4⋊Q8, Dic10 [×2], C22×D5, C23.81C23, C2×Dic10, C4○D20 [×2], D4×D5 [×2], 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
(1 244)(2 245)(3 246)(4 247)(5 248)(6 249)(7 250)(8 251)(9 252)(10 253)(11 254)(12 255)(13 256)(14 257)(15 258)(16 259)(17 260)(18 241)(19 242)(20 243)(21 164)(22 165)(23 166)(24 167)(25 168)(26 169)(27 170)(28 171)(29 172)(30 173)(31 174)(32 175)(33 176)(34 177)(35 178)(36 179)(37 180)(38 161)(39 162)(40 163)(41 220)(42 201)(43 202)(44 203)(45 204)(46 205)(47 206)(48 207)(49 208)(50 209)(51 210)(52 211)(53 212)(54 213)(55 214)(56 215)(57 216)(58 217)(59 218)(60 219)(61 190)(62 191)(63 192)(64 193)(65 194)(66 195)(67 196)(68 197)(69 198)(70 199)(71 200)(72 181)(73 182)(74 183)(75 184)(76 185)(77 186)(78 187)(79 188)(80 189)(81 108)(82 109)(83 110)(84 111)(85 112)(86 113)(87 114)(88 115)(89 116)(90 117)(91 118)(92 119)(93 120)(94 101)(95 102)(96 103)(97 104)(98 105)(99 106)(100 107)(121 285)(122 286)(123 287)(124 288)(125 289)(126 290)(127 291)(128 292)(129 293)(130 294)(131 295)(132 296)(133 297)(134 298)(135 299)(136 300)(137 281)(138 282)(139 283)(140 284)(141 236)(142 237)(143 238)(144 239)(145 240)(146 221)(147 222)(148 223)(149 224)(150 225)(151 226)(152 227)(153 228)(154 229)(155 230)(156 231)(157 232)(158 233)(159 234)(160 235)(261 313)(262 314)(263 315)(264 316)(265 317)(266 318)(267 319)(268 320)(269 301)(270 302)(271 303)(272 304)(273 305)(274 306)(275 307)(276 308)(277 309)(278 310)(279 311)(280 312)
(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)
(1 127 94 29)(2 282 95 163)(3 129 96 31)(4 284 97 165)(5 131 98 33)(6 286 99 167)(7 133 100 35)(8 288 81 169)(9 135 82 37)(10 290 83 171)(11 137 84 39)(12 292 85 173)(13 139 86 21)(14 294 87 175)(15 121 88 23)(16 296 89 177)(17 123 90 25)(18 298 91 179)(19 125 92 27)(20 300 93 161)(22 247 140 104)(24 249 122 106)(26 251 124 108)(28 253 126 110)(30 255 128 112)(32 257 130 114)(34 259 132 116)(36 241 134 118)(38 243 136 120)(40 245 138 102)(41 314 200 232)(42 273 181 148)(43 316 182 234)(44 275 183 150)(45 318 184 236)(46 277 185 152)(47 320 186 238)(48 279 187 154)(49 302 188 240)(50 261 189 156)(51 304 190 222)(52 263 191 158)(53 306 192 224)(54 265 193 160)(55 308 194 226)(56 267 195 142)(57 310 196 228)(58 269 197 144)(59 312 198 230)(60 271 199 146)(61 147 210 272)(62 233 211 315)(63 149 212 274)(64 235 213 317)(65 151 214 276)(66 237 215 319)(67 153 216 278)(68 239 217 301)(69 155 218 280)(70 221 219 303)(71 157 220 262)(72 223 201 305)(73 159 202 264)(74 225 203 307)(75 141 204 266)(76 227 205 309)(77 143 206 268)(78 229 207 311)(79 145 208 270)(80 231 209 313)(101 172 244 291)(103 174 246 293)(105 176 248 295)(107 178 250 297)(109 180 252 299)(111 162 254 281)(113 164 256 283)(115 166 258 285)(117 168 260 287)(119 170 242 289)
(1 238 94 320)(2 152 95 277)(3 236 96 318)(4 150 97 275)(5 234 98 316)(6 148 99 273)(7 232 100 314)(8 146 81 271)(9 230 82 312)(10 144 83 269)(11 228 84 310)(12 142 85 267)(13 226 86 308)(14 160 87 265)(15 224 88 306)(16 158 89 263)(17 222 90 304)(18 156 91 261)(19 240 92 302)(20 154 93 279)(21 214 139 65)(22 44 140 183)(23 212 121 63)(24 42 122 181)(25 210 123 61)(26 60 124 199)(27 208 125 79)(28 58 126 197)(29 206 127 77)(30 56 128 195)(31 204 129 75)(32 54 130 193)(33 202 131 73)(34 52 132 191)(35 220 133 71)(36 50 134 189)(37 218 135 69)(38 48 136 187)(39 216 137 67)(40 46 138 185)(41 297 200 178)(43 295 182 176)(45 293 184 174)(47 291 186 172)(49 289 188 170)(51 287 190 168)(53 285 192 166)(55 283 194 164)(57 281 196 162)(59 299 198 180)(62 177 211 296)(64 175 213 294)(66 173 215 292)(68 171 217 290)(70 169 219 288)(72 167 201 286)(74 165 203 284)(76 163 205 282)(78 161 207 300)(80 179 209 298)(101 268 244 143)(102 309 245 227)(103 266 246 141)(104 307 247 225)(105 264 248 159)(106 305 249 223)(107 262 250 157)(108 303 251 221)(109 280 252 155)(110 301 253 239)(111 278 254 153)(112 319 255 237)(113 276 256 151)(114 317 257 235)(115 274 258 149)(116 315 259 233)(117 272 260 147)(118 313 241 231)(119 270 242 145)(120 311 243 229)

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

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

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

62 conjugacy classes

class 1 2A···2G4A···4F4G···4N5A5B10A···10N20A···20X
order12···24···44···45510···1020···20
size11···14···420···20222···24···4

62 irreducible representations

dim1111122222222444
type++++++--++-+--
imageC1C2C2C2C2D4Q8Q8D5C4○D4D10Dic10C4○D20D4×D5D42D5Q8×D5
kernelC2.(C20⋊Q8)C10.10C42C5×C2.C42C2×C10.D4C2×C4⋊Dic5C2×Dic5C2×Dic5C2×C20C2.C42C2×C10C22×C4C2×C4C22C22C22C22
# reps12131422266816422

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

4000000
0400000
0040000
0004000
0000400
0000040
,
090000
900000
00391100
00142500
0000236
00003518
,
010000
100000
009000
000900
000001
000010
,
12150000
26290000
00101300
0083100
0000132
00003928

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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