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G = C2×C20⋊2Q8  order 320 = 26·5

Direct product of C2 and C20⋊2Q8

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C20⋊2Q8, C42.272D10, C20⋊5(C2×Q8), C10⋊1(C4⋊Q8), (C2×C20)⋊12Q8, (C2×C4)⋊9Dic10, C4⋊3(C2×Dic10), C4.42(C2×D20), (C2×C4).97D20, (C2×C20).388D4, C20.285(C2×D4), C10.1(C22×D4), (C2×C42).19D5, C2.4(C22×D20), C10.2(C22×Q8), (C2×C10).12C24, C22.62(C2×D20), (C4×C20).312C22, (C2×C20).778C23, (C22×C4).434D10, (C2×Dic5).1C23, C2.4(C22×Dic10), C22.59(C23×D5), C4⋊Dic5.286C22, (C22×Dic10).8C2, C22.34(C2×Dic10), C23.310(C22×D5), (C22×C20).521C22, (C22×C10).374C23, (C2×Dic10).230C22, (C22×Dic5).72C22, C5⋊1(C2×C4⋊Q8), (C2×C4×C20).13C2, (C2×C10).46(C2×Q8), (C2×C10).168(C2×D4), (C2×C4⋊Dic5).25C2, (C2×C4).727(C22×D5), SmallGroup(320,1140)

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

Subgroups: 894 in 290 conjugacy classes, 159 normal (13 characteristic)
C1, C2, C2 [×6], C4 [×12], C4 [×8], C22, C22 [×6], C5, C2×C4 [×18], C2×C4 [×16], Q8 [×16], C23, C10, C10 [×6], C42 [×4], C4⋊C4 [×16], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×16], Dic5 [×8], C20 [×12], C2×C10, C2×C10 [×6], C2×C42, C2×C4⋊C4 [×4], C4⋊Q8 [×8], C22×Q8 [×2], Dic10 [×16], C2×Dic5 [×8], C2×Dic5 [×8], C2×C20 [×18], C22×C10, C2×C4⋊Q8, C4⋊Dic5 [×16], C4×C20 [×4], C2×Dic10 [×8], C2×Dic10 [×8], C22×Dic5 [×4], C22×C20, C22×C20 [×2], C20⋊2Q8 [×8], C2×C4⋊Dic5 [×4], C2×C4×C20, C22×Dic10 [×2], C2×C20⋊2Q8

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], D5, C2×D4 [×6], C2×Q8 [×12], C24, D10 [×7], C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], Dic10 [×8], D20 [×4], C22×D5 [×7], C2×C4⋊Q8, C2×Dic10 [×12], C2×D20 [×6], C23×D5, C20⋊2Q8 [×4], C22×Dic10 [×2], C22×D20, C2×C20⋊2Q8

Generators and relations
 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 >

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

(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)

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

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

Matrix representation ►G ⊆ GL5(𝔽41)

400000
01000
00100
000400
000040
,
400000
00100
040000
000140
000366
,
400000
01000
00100
0003932
000372
,
10000
0252700
0271600
0002727
0001714

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] >;
 

92 conjugacy classes

class 1 2A···2G4A···4L4M···4T5A5B10A···10N20A···20AV
order12···24···44···45510···1020···20
size11···12···220···20222···22···2

92 irreducible representations

dim111112222222
type++++++-+++-+
imageC1C2C2C2C2D4Q8D5D10D10Dic10D20
kernelC2×C20⋊2Q8C20⋊2Q8C2×C4⋊Dic5C2×C4×C20C22×Dic10C2×C20C2×C20C2×C42C42C22×C4C2×C4C2×C4
# reps18412482863216

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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