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## G = C22×C5⋊Q16order 320 = 26·5

### Direct product of C22 and C5⋊Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C22×C5⋊Q16
 Chief series C1 — C5 — C10 — C20 — Dic10 — C2×Dic10 — C22×Dic10 — C22×C5⋊Q16
 Lower central C5 — C10 — C20 — C22×C5⋊Q16
 Upper central C1 — C23 — C22×C4 — C22×Q8

Generators and relations for C22×C5⋊Q16
G = < a,b,c,d,e | a2=b2=c5=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, ede-1=d-1 >

Subgroups: 702 in 258 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2, C4, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, Q8, C23, C10, C10, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, Dic5, C20, C20, C20, C2×C10, C22×C8, C2×Q16, C22×Q8, C22×Q8, C52C8, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×C10, C22×Q16, C2×C52C8, C5⋊Q16, C2×Dic10, C2×Dic10, C22×Dic5, C22×C20, C22×C20, Q8×C10, Q8×C10, C22×C52C8, C2×C5⋊Q16, C22×Dic10, Q8×C2×C10, C22×C5⋊Q16
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2×D4, C24, D10, C2×Q16, C22×D4, C5⋊D4, C22×D5, C22×Q16, C5⋊Q16, C2×C5⋊D4, C23×D5, C2×C5⋊Q16, C22×C5⋊D4, C22×C5⋊Q16

Smallest permutation representation of C22×C5⋊Q16
Regular action on 320 points
Generators in S320
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 33)(9 288)(10 281)(11 282)(12 283)(13 284)(14 285)(15 286)(16 287)(17 183)(18 184)(19 177)(20 178)(21 179)(22 180)(23 181)(24 182)(25 69)(26 70)(27 71)(28 72)(29 65)(30 66)(31 67)(32 68)(41 156)(42 157)(43 158)(44 159)(45 160)(46 153)(47 154)(48 155)(49 146)(50 147)(51 148)(52 149)(53 150)(54 151)(55 152)(56 145)(57 249)(58 250)(59 251)(60 252)(61 253)(62 254)(63 255)(64 256)(73 194)(74 195)(75 196)(76 197)(77 198)(78 199)(79 200)(80 193)(81 125)(82 126)(83 127)(84 128)(85 121)(86 122)(87 123)(88 124)(89 133)(90 134)(91 135)(92 136)(93 129)(94 130)(95 131)(96 132)(97 141)(98 142)(99 143)(100 144)(101 137)(102 138)(103 139)(104 140)(105 219)(106 220)(107 221)(108 222)(109 223)(110 224)(111 217)(112 218)(113 237)(114 238)(115 239)(116 240)(117 233)(118 234)(119 235)(120 236)(161 205)(162 206)(163 207)(164 208)(165 201)(166 202)(167 203)(168 204)(169 213)(170 214)(171 215)(172 216)(173 209)(174 210)(175 211)(176 212)(185 242)(186 243)(187 244)(188 245)(189 246)(190 247)(191 248)(192 241)(225 294)(226 295)(227 296)(228 289)(229 290)(230 291)(231 292)(232 293)(257 301)(258 302)(259 303)(260 304)(261 297)(262 298)(263 299)(264 300)(265 309)(266 310)(267 311)(268 312)(269 305)(270 306)(271 307)(272 308)(273 317)(274 318)(275 319)(276 320)(277 313)(278 314)(279 315)(280 316)
(1 47)(2 48)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 311)(10 312)(11 305)(12 306)(13 307)(14 308)(15 309)(16 310)(17 197)(18 198)(19 199)(20 200)(21 193)(22 194)(23 195)(24 196)(25 145)(26 146)(27 147)(28 148)(29 149)(30 150)(31 151)(32 152)(33 153)(34 154)(35 155)(36 156)(37 157)(38 158)(39 159)(40 160)(49 70)(50 71)(51 72)(52 65)(53 66)(54 67)(55 68)(56 69)(57 173)(58 174)(59 175)(60 176)(61 169)(62 170)(63 171)(64 172)(73 180)(74 181)(75 182)(76 183)(77 184)(78 177)(79 178)(80 179)(81 240)(82 233)(83 234)(84 235)(85 236)(86 237)(87 238)(88 239)(89 220)(90 221)(91 222)(92 223)(93 224)(94 217)(95 218)(96 219)(97 228)(98 229)(99 230)(100 231)(101 232)(102 225)(103 226)(104 227)(105 132)(106 133)(107 134)(108 135)(109 136)(110 129)(111 130)(112 131)(113 122)(114 123)(115 124)(116 125)(117 126)(118 127)(119 128)(120 121)(137 293)(138 294)(139 295)(140 296)(141 289)(142 290)(143 291)(144 292)(161 188)(162 189)(163 190)(164 191)(165 192)(166 185)(167 186)(168 187)(201 241)(202 242)(203 243)(204 244)(205 245)(206 246)(207 247)(208 248)(209 249)(210 250)(211 251)(212 252)(213 253)(214 254)(215 255)(216 256)(257 275)(258 276)(259 277)(260 278)(261 279)(262 280)(263 273)(264 274)(265 286)(266 287)(267 288)(268 281)(269 282)(270 283)(271 284)(272 285)(297 315)(298 316)(299 317)(300 318)(301 319)(302 320)(303 313)(304 314)
(1 110 79 287 291)(2 292 288 80 111)(3 112 73 281 293)(4 294 282 74 105)(5 106 75 283 295)(6 296 284 76 107)(7 108 77 285 289)(8 290 286 78 109)(9 193 217 35 231)(10 232 36 218 194)(11 195 219 37 225)(12 226 38 220 196)(13 197 221 39 227)(14 228 40 222 198)(15 199 223 33 229)(16 230 34 224 200)(17 90 159 104 307)(18 308 97 160 91)(19 92 153 98 309)(20 310 99 154 93)(21 94 155 100 311)(22 312 101 156 95)(23 96 157 102 305)(24 306 103 158 89)(25 313 206 214 239)(26 240 215 207 314)(27 315 208 216 233)(28 234 209 201 316)(29 317 202 210 235)(30 236 211 203 318)(31 319 204 212 237)(32 238 213 205 320)(41 131 180 268 137)(42 138 269 181 132)(43 133 182 270 139)(44 140 271 183 134)(45 135 184 272 141)(46 142 265 177 136)(47 129 178 266 143)(48 144 267 179 130)(49 125 63 190 260)(50 261 191 64 126)(51 127 57 192 262)(52 263 185 58 128)(53 121 59 186 264)(54 257 187 60 122)(55 123 61 188 258)(56 259 189 62 124)(65 273 166 174 119)(66 120 175 167 274)(67 275 168 176 113)(68 114 169 161 276)(69 277 162 170 115)(70 116 171 163 278)(71 279 164 172 117)(72 118 173 165 280)(81 255 247 304 146)(82 147 297 248 256)(83 249 241 298 148)(84 149 299 242 250)(85 251 243 300 150)(86 151 301 244 252)(87 253 245 302 152)(88 145 303 246 254)
(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 84 5 88)(2 83 6 87)(3 82 7 86)(4 81 8 85)(9 192 13 188)(10 191 14 187)(11 190 15 186)(12 189 16 185)(17 276 21 280)(18 275 22 279)(19 274 23 278)(20 273 24 277)(25 129 29 133)(26 136 30 132)(27 135 31 131)(28 134 32 130)(33 121 37 125)(34 128 38 124)(35 127 39 123)(36 126 40 122)(41 233 45 237)(42 240 46 236)(43 239 47 235)(44 238 48 234)(49 223 53 219)(50 222 54 218)(51 221 55 217)(52 220 56 224)(57 227 61 231)(58 226 62 230)(59 225 63 229)(60 232 64 228)(65 89 69 93)(66 96 70 92)(67 95 71 91)(68 94 72 90)(73 297 77 301)(74 304 78 300)(75 303 79 299)(76 302 80 298)(97 176 101 172)(98 175 102 171)(99 174 103 170)(100 173 104 169)(105 146 109 150)(106 145 110 149)(107 152 111 148)(108 151 112 147)(113 156 117 160)(114 155 118 159)(115 154 119 158)(116 153 120 157)(137 216 141 212)(138 215 142 211)(139 214 143 210)(140 213 144 209)(161 311 165 307)(162 310 166 306)(163 309 167 305)(164 308 168 312)(177 318 181 314)(178 317 182 313)(179 316 183 320)(180 315 184 319)(193 262 197 258)(194 261 198 257)(195 260 199 264)(196 259 200 263)(201 271 205 267)(202 270 206 266)(203 269 207 265)(204 268 208 272)(241 284 245 288)(242 283 246 287)(243 282 247 286)(244 281 248 285)(249 296 253 292)(250 295 254 291)(251 294 255 290)(252 293 256 289)

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 8A ··· 8H 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 4 4 4 4 4 4 4 4 4 4 4 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 2 2 2 4 4 4 4 20 20 20 20 2 2 10 ··· 10 2 ··· 2 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + - + + - image C1 C2 C2 C2 C2 D4 D4 D5 Q16 D10 D10 C5⋊D4 C5⋊D4 C5⋊Q16 kernel C22×C5⋊Q16 C22×C5⋊2C8 C2×C5⋊Q16 C22×Dic10 Q8×C2×C10 C2×C20 C22×C10 C22×Q8 C2×C10 C22×C4 C2×Q8 C2×C4 C23 C22 # reps 1 1 12 1 1 3 1 2 8 2 12 12 4 8

Matrix representation of C22×C5⋊Q16 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 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
,
 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 40 0 0 0 0 1 6 0 0 0 0 0 0 6 40 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 40 0 0 0 0 40 0 0 0 0 0 0 0 40 6 0 0 0 0 0 1 0 0 0 0 0 0 12 29 0 0 0 0 12 12
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 11 40 0 0 0 0 40 30

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[0,1,0,0,0,0,40,6,0,0,0,0,0,0,6,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,6,1,0,0,0,0,0,0,12,12,0,0,0,0,29,12],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,40,0,0,0,0,40,30] >;

C22×C5⋊Q16 in GAP, Magma, Sage, TeX

C_2^2\times C_5\rtimes Q_{16}
% in TeX

G:=Group("C2^2xC5:Q16");
// GroupNames label

G:=SmallGroup(320,1481);
// by ID

G=gap.SmallGroup(320,1481);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,675,136,1684,235,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^5=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽