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## G = C5×Q8.Q8order 320 = 26·5

### Direct product of C5 and Q8.Q8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×Q8.Q8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — C5×C4⋊C4 — C5×C42.C2 — C5×Q8.Q8
 Lower central C1 — C2 — C2×C4 — C5×Q8.Q8
 Upper central C1 — C2×C10 — C4×C20 — C5×Q8.Q8

Generators and relations for C5×Q8.Q8
G = < a,b,c,d,e | a5=b4=d4=1, c2=b2, e2=b2d2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, cd=dc, ece-1=b-1c, ede-1=b2d-1 >

Subgroups: 138 in 90 conjugacy classes, 54 normal (50 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, Q8, C10, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C20, C20, C2×C10, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×Q8, C42.C2, C40, C2×C20, C2×C20, C5×Q8, C5×Q8, Q8.Q8, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C40, Q8×C10, C5×Q8⋊C4, C5×C4⋊C8, C5×C4.Q8, C5×C2.D8, Q8×C20, C5×C42.C2, C5×Q8.Q8
Quotients: C1, C2, C22, C5, D4, Q8, C23, C10, C2×D4, C2×Q8, C4○D4, C2×C10, C22⋊Q8, C4○D8, C8.C22, C5×D4, C5×Q8, C22×C10, Q8.Q8, D4×C10, Q8×C10, C5×C4○D4, C5×C22⋊Q8, C5×C4○D8, C5×C8.C22, C5×Q8.Q8

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

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

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

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

95 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E ··· 4I 4J 4K 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 20A ··· 20P 20Q ··· 20AJ 20AK ··· 20AR 40A ··· 40P order 1 2 2 2 4 4 4 4 4 ··· 4 4 4 5 5 5 5 8 8 8 8 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 2 2 4 ··· 4 8 8 1 1 1 1 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 8 ··· 8 4 ··· 4

95 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 C10 D4 Q8 C4○D4 C4○D8 C5×D4 C5×Q8 C5×C4○D4 C5×C4○D8 C8.C22 C5×C8.C22 kernel C5×Q8.Q8 C5×Q8⋊C4 C5×C4⋊C8 C5×C4.Q8 C5×C2.D8 Q8×C20 C5×C42.C2 Q8.Q8 Q8⋊C4 C4⋊C8 C4.Q8 C2.D8 C4×Q8 C42.C2 C2×C20 C5×Q8 C20 C10 C2×C4 Q8 C4 C2 C10 C2 # reps 1 2 1 1 1 1 1 4 8 4 4 4 4 4 2 2 2 4 8 8 8 16 1 4

Matrix representation of C5×Q8.Q8 in GL4(𝔽41) generated by

 10 0 0 0 0 10 0 0 0 0 18 0 0 0 0 18
,
 0 1 0 0 40 0 0 0 0 0 40 0 0 0 0 40
,
 21 38 0 0 38 20 0 0 0 0 0 1 0 0 1 0
,
 32 0 0 0 0 32 0 0 0 0 0 32 0 0 32 0
,
 17 32 0 0 32 24 0 0 0 0 0 40 0 0 1 0
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,18,0,0,0,0,18],[0,40,0,0,1,0,0,0,0,0,40,0,0,0,0,40],[21,38,0,0,38,20,0,0,0,0,0,1,0,0,1,0],[32,0,0,0,0,32,0,0,0,0,0,32,0,0,32,0],[17,32,0,0,32,24,0,0,0,0,0,1,0,0,40,0] >;

C5×Q8.Q8 in GAP, Magma, Sage, TeX

C_5\times Q_8.Q_8
% in TeX

G:=Group("C5xQ8.Q8");
// GroupNames label

G:=SmallGroup(320,980);
// by ID

G=gap.SmallGroup(320,980);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,280,589,1408,1766,856,10085,2539,124]);
// Polycyclic

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

׿
×
𝔽