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G = Q8×C2×C20order 320 = 26·5

Direct product of C2×C20 and Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: Q8×C2×C20, C2.5(C23×C20), C4.17(C22×C20), (C2×C42).17C10, C10.78(C23×C4), C42.86(C2×C10), C22.17(Q8×C10), C10.56(C22×Q8), (C2×C20).708C23, (C4×C20).370C22, C20.221(C22×C4), (C2×C10).336C24, C22.9(C23×C10), (C22×Q8).10C10, C22.26(C22×C20), C23.68(C22×C10), (Q8×C10).282C22, (C22×C10).468C23, (C22×C20).594C22, C2.2(Q8×C2×C10), (C2×C4×C20).40C2, C2.3(C10×C4○D4), (C2×C4⋊C4).22C10, (C10×C4⋊C4).51C2, (Q8×C2×C10).20C2, C4⋊C4.80(C2×C10), (C2×C4).52(C2×C20), (C2×C20).446(C2×C4), C10.222(C2×C4○D4), (C2×Q8).70(C2×C10), (C2×C10).115(C2×Q8), C22.28(C5×C4○D4), (C5×C4⋊C4).405C22, (C2×C4).55(C22×C10), (C22×C4).98(C2×C10), (C2×C10).228(C4○D4), (C2×C10).347(C22×C4), SmallGroup(320,1518)

Series: Derived Chief Lower central Upper central

C1C2 — Q8×C2×C20
C1C2C22C2×C10C2×C20C5×C4⋊C4Q8×C20 — Q8×C2×C20
C1C2 — Q8×C2×C20
C1C22×C20 — Q8×C2×C20

Generators and relations for Q8×C2×C20
 G = < a,b,c,d | a2=b20=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 322 in 298 conjugacy classes, 274 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×16], C4 [×6], C22, C22 [×6], C5, C2×C4 [×30], C2×C4 [×6], Q8 [×16], C23, C10 [×3], C10 [×4], C42 [×12], C4⋊C4 [×12], C22×C4, C22×C4 [×6], C2×Q8 [×12], C20 [×16], C20 [×6], C2×C10, C2×C10 [×6], C2×C42 [×3], C2×C4⋊C4 [×3], C4×Q8 [×8], C22×Q8, C2×C20 [×30], C2×C20 [×6], C5×Q8 [×16], C22×C10, C2×C4×Q8, C4×C20 [×12], C5×C4⋊C4 [×12], C22×C20, C22×C20 [×6], Q8×C10 [×12], C2×C4×C20 [×3], C10×C4⋊C4 [×3], Q8×C20 [×8], Q8×C2×C10, Q8×C2×C20
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C5, C2×C4 [×28], Q8 [×4], C23 [×15], C10 [×15], C22×C4 [×14], C2×Q8 [×6], C4○D4 [×2], C24, C20 [×8], C2×C10 [×35], C4×Q8 [×4], C23×C4, C22×Q8, C2×C4○D4, C2×C20 [×28], C5×Q8 [×4], C22×C10 [×15], C2×C4×Q8, C22×C20 [×14], Q8×C10 [×6], C5×C4○D4 [×2], C23×C10, Q8×C20 [×4], C23×C20, Q8×C2×C10, C10×C4○D4, Q8×C2×C20

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

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

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

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

200 conjugacy classes

class 1 2A···2G4A···4H4I···4AF5A5B5C5D10A···10AB20A···20AF20AG···20DX
order12···24···44···4555510···1020···2020···20
size11···11···12···211111···11···12···2

200 irreducible representations

dim1111111111112222
type+++++-
imageC1C2C2C2C2C4C5C10C10C10C10C20Q8C4○D4C5×Q8C5×C4○D4
kernelQ8×C2×C20C2×C4×C20C10×C4⋊C4Q8×C20Q8×C2×C10Q8×C10C2×C4×Q8C2×C42C2×C4⋊C4C4×Q8C22×Q8C2×Q8C2×C20C2×C10C2×C4C22
# reps13381164121232464441616

Matrix representation of Q8×C2×C20 in GL4(𝔽41) generated by

1000
04000
00400
00040
,
9000
04000
00100
00010
,
40000
04000
0001
00400
,
1000
04000
0090
00032
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[9,0,0,0,0,40,0,0,0,0,10,0,0,0,0,10],[40,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[1,0,0,0,0,40,0,0,0,0,9,0,0,0,0,32] >;

Q8×C2×C20 in GAP, Magma, Sage, TeX

Q_8\times C_2\times C_{20}
% in TeX

G:=Group("Q8xC2xC20");
// GroupNames label

G:=SmallGroup(320,1518);
// by ID

G=gap.SmallGroup(320,1518);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149,568,1276]);
// 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,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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