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## G = C5×Q82order 320 = 26·5

### Direct product of C5, Q8 and Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×Q82
 Chief series C1 — C2 — C22 — C2×C10 — C2×C20 — C5×C4⋊C4 — C5×C4⋊Q8 — C5×Q82
 Lower central C1 — C22 — C5×Q82
 Upper central C1 — C2×C10 — C5×Q82

Generators and relations for C5×Q82
G = < a,b,c,d,e | a5=b4=d4=1, c2=b2, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 266 in 212 conjugacy classes, 182 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C5, C2×C4, Q8, Q8, C10, C10, C42, C4⋊C4, C2×Q8, C20, C20, C2×C10, C4×Q8, C4⋊Q8, C2×C20, C5×Q8, C5×Q8, Q82, C4×C20, C5×C4⋊C4, Q8×C10, Q8×C20, C5×C4⋊Q8, C5×Q82
Quotients: C1, C2, C22, C5, Q8, C23, C10, C2×Q8, C24, C2×C10, C22×Q8, 2+ 1+4, C5×Q8, C22×C10, Q82, Q8×C10, C23×C10, Q8×C2×C10, C5×2+ 1+4, C5×Q82

Smallest permutation representation of C5×Q82
Regular action on 320 points
Generators in S320
(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 63 30 75)(2 64 26 71)(3 65 27 72)(4 61 28 73)(5 62 29 74)(6 295 316 283)(7 291 317 284)(8 292 318 285)(9 293 319 281)(10 294 320 282)(11 290 16 276)(12 286 17 277)(13 287 18 278)(14 288 19 279)(15 289 20 280)(21 59 33 68)(22 60 34 69)(23 56 35 70)(24 57 31 66)(25 58 32 67)(36 83 50 95)(37 84 46 91)(38 85 47 92)(39 81 48 93)(40 82 49 94)(41 79 53 88)(42 80 54 89)(43 76 55 90)(44 77 51 86)(45 78 52 87)(96 143 110 155)(97 144 106 151)(98 145 107 152)(99 141 108 153)(100 142 109 154)(101 139 113 148)(102 140 114 149)(103 136 115 150)(104 137 111 146)(105 138 112 147)(116 163 130 175)(117 164 126 171)(118 165 127 172)(119 161 128 173)(120 162 129 174)(121 159 133 168)(122 160 134 169)(123 156 135 170)(124 157 131 166)(125 158 132 167)(176 235 190 223)(177 231 186 224)(178 232 187 225)(179 233 188 221)(180 234 189 222)(181 228 193 219)(182 229 194 220)(183 230 195 216)(184 226 191 217)(185 227 192 218)(196 255 210 243)(197 251 206 244)(198 252 207 245)(199 253 208 241)(200 254 209 242)(201 248 213 239)(202 249 214 240)(203 250 215 236)(204 246 211 237)(205 247 212 238)(256 315 270 303)(257 311 266 304)(258 312 267 305)(259 313 268 301)(260 314 269 302)(261 308 273 299)(262 309 274 300)(263 310 275 296)(264 306 271 297)(265 307 272 298)
(1 203 30 215)(2 204 26 211)(3 205 27 212)(4 201 28 213)(5 202 29 214)(6 150 316 136)(7 146 317 137)(8 147 318 138)(9 148 319 139)(10 149 320 140)(11 155 16 143)(12 151 17 144)(13 152 18 145)(14 153 19 141)(15 154 20 142)(21 199 33 208)(22 200 34 209)(23 196 35 210)(24 197 31 206)(25 198 32 207)(36 190 50 176)(37 186 46 177)(38 187 47 178)(39 188 48 179)(40 189 49 180)(41 193 53 181)(42 194 54 182)(43 195 55 183)(44 191 51 184)(45 192 52 185)(56 243 70 255)(57 244 66 251)(58 245 67 252)(59 241 68 253)(60 242 69 254)(61 239 73 248)(62 240 74 249)(63 236 75 250)(64 237 71 246)(65 238 72 247)(76 230 90 216)(77 226 86 217)(78 227 87 218)(79 228 88 219)(80 229 89 220)(81 233 93 221)(82 234 94 222)(83 235 95 223)(84 231 91 224)(85 232 92 225)(96 290 110 276)(97 286 106 277)(98 287 107 278)(99 288 108 279)(100 289 109 280)(101 293 113 281)(102 294 114 282)(103 295 115 283)(104 291 111 284)(105 292 112 285)(116 263 130 275)(117 264 126 271)(118 265 127 272)(119 261 128 273)(120 262 129 274)(121 259 133 268)(122 260 134 269)(123 256 135 270)(124 257 131 266)(125 258 132 267)(156 303 170 315)(157 304 166 311)(158 305 167 312)(159 301 168 313)(160 302 169 314)(161 299 173 308)(162 300 174 309)(163 296 175 310)(164 297 171 306)(165 298 172 307)
(1 55 35 36)(2 51 31 37)(3 52 32 38)(4 53 33 39)(5 54 34 40)(6 303 16 310)(7 304 17 306)(8 305 18 307)(9 301 19 308)(10 302 20 309)(11 296 316 315)(12 297 317 311)(13 298 318 312)(14 299 319 313)(15 300 320 314)(21 48 28 41)(22 49 29 42)(23 50 30 43)(24 46 26 44)(25 47 27 45)(56 95 75 76)(57 91 71 77)(58 92 72 78)(59 93 73 79)(60 94 74 80)(61 88 68 81)(62 89 69 82)(63 90 70 83)(64 86 66 84)(65 87 67 85)(96 116 115 135)(97 117 111 131)(98 118 112 132)(99 119 113 133)(100 120 114 134)(101 121 108 128)(102 122 109 129)(103 123 110 130)(104 124 106 126)(105 125 107 127)(136 156 155 175)(137 157 151 171)(138 158 152 172)(139 159 153 173)(140 160 154 174)(141 161 148 168)(142 162 149 169)(143 163 150 170)(144 164 146 166)(145 165 147 167)(176 215 195 196)(177 211 191 197)(178 212 192 198)(179 213 193 199)(180 214 194 200)(181 208 188 201)(182 209 189 202)(183 210 190 203)(184 206 186 204)(185 207 187 205)(216 255 235 236)(217 251 231 237)(218 252 232 238)(219 253 233 239)(220 254 234 240)(221 248 228 241)(222 249 229 242)(223 250 230 243)(224 246 226 244)(225 247 227 245)(256 276 275 295)(257 277 271 291)(258 278 272 292)(259 279 273 293)(260 280 274 294)(261 281 268 288)(262 282 269 289)(263 283 270 290)(264 284 266 286)(265 285 267 287)
(1 115 35 96)(2 111 31 97)(3 112 32 98)(4 113 33 99)(5 114 34 100)(6 243 16 250)(7 244 17 246)(8 245 18 247)(9 241 19 248)(10 242 20 249)(11 236 316 255)(12 237 317 251)(13 238 318 252)(14 239 319 253)(15 240 320 254)(21 108 28 101)(22 109 29 102)(23 110 30 103)(24 106 26 104)(25 107 27 105)(36 135 55 116)(37 131 51 117)(38 132 52 118)(39 133 53 119)(40 134 54 120)(41 128 48 121)(42 129 49 122)(43 130 50 123)(44 126 46 124)(45 127 47 125)(56 155 75 136)(57 151 71 137)(58 152 72 138)(59 153 73 139)(60 154 74 140)(61 148 68 141)(62 149 69 142)(63 150 70 143)(64 146 66 144)(65 147 67 145)(76 175 95 156)(77 171 91 157)(78 172 92 158)(79 173 93 159)(80 174 94 160)(81 168 88 161)(82 169 89 162)(83 170 90 163)(84 166 86 164)(85 167 87 165)(176 256 195 275)(177 257 191 271)(178 258 192 272)(179 259 193 273)(180 260 194 274)(181 261 188 268)(182 262 189 269)(183 263 190 270)(184 264 186 266)(185 265 187 267)(196 276 215 295)(197 277 211 291)(198 278 212 292)(199 279 213 293)(200 280 214 294)(201 281 208 288)(202 282 209 289)(203 283 210 290)(204 284 206 286)(205 285 207 287)(216 296 235 315)(217 297 231 311)(218 298 232 312)(219 299 233 313)(220 300 234 314)(221 301 228 308)(222 302 229 309)(223 303 230 310)(224 304 226 306)(225 305 227 307)

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

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

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

125 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4L 4M ··· 4U 5A 5B 5C 5D 10A ··· 10L 20A ··· 20AV 20AW ··· 20CF order 1 2 2 2 4 ··· 4 4 ··· 4 5 5 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 ··· 2 4 ··· 4 1 1 1 1 1 ··· 1 2 ··· 2 4 ··· 4

125 irreducible representations

 dim 1 1 1 1 1 1 2 2 4 4 type + + + - + image C1 C2 C2 C5 C10 C10 Q8 C5×Q8 2+ 1+4 C5×2+ 1+4 kernel C5×Q82 Q8×C20 C5×C4⋊Q8 Q82 C4×Q8 C4⋊Q8 C5×Q8 Q8 C10 C2 # reps 1 6 9 4 24 36 8 32 1 4

Matrix representation of C5×Q82 in GL4(𝔽41) generated by

 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 40 39 0 0 1 1 0 0 0 0 1 0 0 0 0 1
,
 34 14 0 0 14 7 0 0 0 0 1 0 0 0 0 1
,
 40 0 0 0 0 40 0 0 0 0 1 39 0 0 1 40
,
 40 0 0 0 0 40 0 0 0 0 4 4 0 0 6 37
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[40,1,0,0,39,1,0,0,0,0,1,0,0,0,0,1],[34,14,0,0,14,7,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,1,0,0,39,40],[40,0,0,0,0,40,0,0,0,0,4,6,0,0,4,37] >;

C5×Q82 in GAP, Magma, Sage, TeX

C_5\times Q_8^2
% in TeX

G:=Group("C5xQ8^2");
// GroupNames label

G:=SmallGroup(320,1560);
// by ID

G=gap.SmallGroup(320,1560);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1680,1149,568,3446,856,1242,304]);
// Polycyclic

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

׿
×
𝔽