direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×Q8⋊Q8, C20.53SD16, (C5×Q8)⋊8Q8, Q8⋊1(C5×Q8), C4⋊C8.9C10, C4⋊Q8.5C10, (C4×Q8).6C10, C4.13(Q8×C10), C4.Q8.4C10, (C2×C20).331D4, (Q8×C20).19C2, C20.119(C2×Q8), C4.12(C5×SD16), C42.21(C2×C10), Q8⋊C4.4C10, C2.10(C10×SD16), C10.90(C2×SD16), C22.96(D4×C10), C20.312(C4○D4), (C2×C20).931C23, (C2×C40).302C22, (C4×C20).263C22, C10.94(C22⋊Q8), (Q8×C10).263C22, C10.140(C8.C22), (C5×C4⋊C8).22C2, C4.24(C5×C4○D4), (C5×C4⋊Q8).20C2, C4⋊C4.12(C2×C10), (C2×C8).39(C2×C10), (C2×C4).132(C5×D4), (C5×C4.Q8).11C2, C2.13(C5×C22⋊Q8), (C2×C10).652(C2×D4), (C2×Q8).50(C2×C10), C2.15(C5×C8.C22), (C5×C4⋊C4).234C22, (C5×Q8⋊C4).13C2, (C2×C4).106(C22×C10), SmallGroup(320,976)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Q8⋊Q8
G = < a,b,c,d,e | a5=b4=d4=1, c2=b2, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, cd=dc, ece-1=b-1c, ede-1=d-1 >
Subgroups: 154 in 96 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C4, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, Q8, C10, C42, C42, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C2×Q8, C20, C20, C20, C2×C10, Q8⋊C4, C4⋊C8, C4.Q8, C4×Q8, C4⋊Q8, C40, C2×C20, C2×C20, C5×Q8, C5×Q8, Q8⋊Q8, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, Q8×C10, Q8×C10, C5×Q8⋊C4, C5×C4⋊C8, C5×C4.Q8, Q8×C20, C5×C4⋊Q8, C5×Q8⋊Q8
Quotients: C1, C2, C22, C5, D4, Q8, C23, C10, SD16, C2×D4, C2×Q8, C4○D4, C2×C10, C22⋊Q8, C2×SD16, C8.C22, C5×D4, C5×Q8, C22×C10, Q8⋊Q8, C5×SD16, D4×C10, Q8×C10, C5×C4○D4, C5×C22⋊Q8, C10×SD16, C5×C8.C22, 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 29 66)(2 38 30 67)(3 39 26 68)(4 40 27 69)(5 36 28 70)(6 316 296 20)(7 317 297 16)(8 318 298 17)(9 319 299 18)(10 320 300 19)(11 46 53 41)(12 47 54 42)(13 48 55 43)(14 49 51 44)(15 50 52 45)(21 311 302 35)(22 312 303 31)(23 313 304 32)(24 314 305 33)(25 315 301 34)(56 76 85 110)(57 77 81 106)(58 78 82 107)(59 79 83 108)(60 80 84 109)(61 75 91 89)(62 71 92 90)(63 72 93 86)(64 73 94 87)(65 74 95 88)(96 150 125 116)(97 146 121 117)(98 147 122 118)(99 148 123 119)(100 149 124 120)(101 129 131 115)(102 130 132 111)(103 126 133 112)(104 127 134 113)(105 128 135 114)(136 156 165 190)(137 157 161 186)(138 158 162 187)(139 159 163 188)(140 160 164 189)(141 155 171 169)(142 151 172 170)(143 152 173 166)(144 153 174 167)(145 154 175 168)(176 196 205 230)(177 197 201 226)(178 198 202 227)(179 199 203 228)(180 200 204 229)(181 195 211 209)(182 191 212 210)(183 192 213 206)(184 193 214 207)(185 194 215 208)(216 270 245 236)(217 266 241 237)(218 267 242 238)(219 268 243 239)(220 269 244 240)(221 249 251 235)(222 250 252 231)(223 246 253 232)(224 247 254 233)(225 248 255 234)(256 276 285 310)(257 277 281 306)(258 278 282 307)(259 279 283 308)(260 280 284 309)(261 275 291 289)(262 271 292 290)(263 272 293 286)(264 273 294 287)(265 274 295 288)
(1 57 29 81)(2 58 30 82)(3 59 26 83)(4 60 27 84)(5 56 28 85)(6 276 296 310)(7 277 297 306)(8 278 298 307)(9 279 299 308)(10 280 300 309)(11 93 53 63)(12 94 54 64)(13 95 55 65)(14 91 51 61)(15 92 52 62)(16 281 317 257)(17 282 318 258)(18 283 319 259)(19 284 320 260)(20 285 316 256)(21 271 302 290)(22 272 303 286)(23 273 304 287)(24 274 305 288)(25 275 301 289)(31 293 312 263)(32 294 313 264)(33 295 314 265)(34 291 315 261)(35 292 311 262)(36 110 70 76)(37 106 66 77)(38 107 67 78)(39 108 68 79)(40 109 69 80)(41 86 46 72)(42 87 47 73)(43 88 48 74)(44 89 49 75)(45 90 50 71)(96 165 125 136)(97 161 121 137)(98 162 122 138)(99 163 123 139)(100 164 124 140)(101 171 131 141)(102 172 132 142)(103 173 133 143)(104 174 134 144)(105 175 135 145)(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)(176 270 205 236)(177 266 201 237)(178 267 202 238)(179 268 203 239)(180 269 204 240)(181 249 211 235)(182 250 212 231)(183 246 213 232)(184 247 214 233)(185 248 215 234)(191 222 210 252)(192 223 206 253)(193 224 207 254)(194 225 208 255)(195 221 209 251)(196 216 230 245)(197 217 226 241)(198 218 227 242)(199 219 228 243)(200 220 229 244)
(1 104 12 121)(2 105 13 122)(3 101 14 123)(4 102 15 124)(5 103 11 125)(6 223 303 245)(7 224 304 241)(8 225 305 242)(9 221 301 243)(10 222 302 244)(16 233 313 266)(17 234 314 267)(18 235 315 268)(19 231 311 269)(20 232 312 270)(21 220 300 252)(22 216 296 253)(23 217 297 254)(24 218 298 255)(25 219 299 251)(26 131 51 99)(27 132 52 100)(28 133 53 96)(29 134 54 97)(30 135 55 98)(31 236 316 246)(32 237 317 247)(33 238 318 248)(34 239 319 249)(35 240 320 250)(36 126 46 116)(37 127 47 117)(38 128 48 118)(39 129 49 119)(40 130 50 120)(41 150 70 112)(42 146 66 113)(43 147 67 114)(44 148 68 115)(45 149 69 111)(56 173 93 136)(57 174 94 137)(58 175 95 138)(59 171 91 139)(60 172 92 140)(61 163 83 141)(62 164 84 142)(63 165 85 143)(64 161 81 144)(65 162 82 145)(71 189 109 151)(72 190 110 152)(73 186 106 153)(74 187 107 154)(75 188 108 155)(76 166 86 156)(77 167 87 157)(78 168 88 158)(79 169 89 159)(80 170 90 160)(176 256 213 293)(177 257 214 294)(178 258 215 295)(179 259 211 291)(180 260 212 292)(181 261 203 283)(182 262 204 284)(183 263 205 285)(184 264 201 281)(185 265 202 282)(191 271 229 309)(192 272 230 310)(193 273 226 306)(194 274 227 307)(195 275 228 308)(196 276 206 286)(197 277 207 287)(198 278 208 288)(199 279 209 289)(200 280 210 290)
(1 184 12 201)(2 185 13 202)(3 181 14 203)(4 182 15 204)(5 183 11 205)(6 165 303 143)(7 161 304 144)(8 162 305 145)(9 163 301 141)(10 164 302 142)(16 186 313 153)(17 187 314 154)(18 188 315 155)(19 189 311 151)(20 190 312 152)(21 172 300 140)(22 173 296 136)(23 174 297 137)(24 175 298 138)(25 171 299 139)(26 211 51 179)(27 212 52 180)(28 213 53 176)(29 214 54 177)(30 215 55 178)(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)(41 230 70 192)(42 226 66 193)(43 227 67 194)(44 228 68 195)(45 229 69 191)(56 253 93 216)(57 254 94 217)(58 255 95 218)(59 251 91 219)(60 252 92 220)(61 243 83 221)(62 244 84 222)(63 245 85 223)(64 241 81 224)(65 242 82 225)(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)(96 293 133 256)(97 294 134 257)(98 295 135 258)(99 291 131 259)(100 292 132 260)(101 283 123 261)(102 284 124 262)(103 285 125 263)(104 281 121 264)(105 282 122 265)(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), (1,37,29,66)(2,38,30,67)(3,39,26,68)(4,40,27,69)(5,36,28,70)(6,316,296,20)(7,317,297,16)(8,318,298,17)(9,319,299,18)(10,320,300,19)(11,46,53,41)(12,47,54,42)(13,48,55,43)(14,49,51,44)(15,50,52,45)(21,311,302,35)(22,312,303,31)(23,313,304,32)(24,314,305,33)(25,315,301,34)(56,76,85,110)(57,77,81,106)(58,78,82,107)(59,79,83,108)(60,80,84,109)(61,75,91,89)(62,71,92,90)(63,72,93,86)(64,73,94,87)(65,74,95,88)(96,150,125,116)(97,146,121,117)(98,147,122,118)(99,148,123,119)(100,149,124,120)(101,129,131,115)(102,130,132,111)(103,126,133,112)(104,127,134,113)(105,128,135,114)(136,156,165,190)(137,157,161,186)(138,158,162,187)(139,159,163,188)(140,160,164,189)(141,155,171,169)(142,151,172,170)(143,152,173,166)(144,153,174,167)(145,154,175,168)(176,196,205,230)(177,197,201,226)(178,198,202,227)(179,199,203,228)(180,200,204,229)(181,195,211,209)(182,191,212,210)(183,192,213,206)(184,193,214,207)(185,194,215,208)(216,270,245,236)(217,266,241,237)(218,267,242,238)(219,268,243,239)(220,269,244,240)(221,249,251,235)(222,250,252,231)(223,246,253,232)(224,247,254,233)(225,248,255,234)(256,276,285,310)(257,277,281,306)(258,278,282,307)(259,279,283,308)(260,280,284,309)(261,275,291,289)(262,271,292,290)(263,272,293,286)(264,273,294,287)(265,274,295,288), (1,57,29,81)(2,58,30,82)(3,59,26,83)(4,60,27,84)(5,56,28,85)(6,276,296,310)(7,277,297,306)(8,278,298,307)(9,279,299,308)(10,280,300,309)(11,93,53,63)(12,94,54,64)(13,95,55,65)(14,91,51,61)(15,92,52,62)(16,281,317,257)(17,282,318,258)(18,283,319,259)(19,284,320,260)(20,285,316,256)(21,271,302,290)(22,272,303,286)(23,273,304,287)(24,274,305,288)(25,275,301,289)(31,293,312,263)(32,294,313,264)(33,295,314,265)(34,291,315,261)(35,292,311,262)(36,110,70,76)(37,106,66,77)(38,107,67,78)(39,108,68,79)(40,109,69,80)(41,86,46,72)(42,87,47,73)(43,88,48,74)(44,89,49,75)(45,90,50,71)(96,165,125,136)(97,161,121,137)(98,162,122,138)(99,163,123,139)(100,164,124,140)(101,171,131,141)(102,172,132,142)(103,173,133,143)(104,174,134,144)(105,175,135,145)(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)(176,270,205,236)(177,266,201,237)(178,267,202,238)(179,268,203,239)(180,269,204,240)(181,249,211,235)(182,250,212,231)(183,246,213,232)(184,247,214,233)(185,248,215,234)(191,222,210,252)(192,223,206,253)(193,224,207,254)(194,225,208,255)(195,221,209,251)(196,216,230,245)(197,217,226,241)(198,218,227,242)(199,219,228,243)(200,220,229,244), (1,104,12,121)(2,105,13,122)(3,101,14,123)(4,102,15,124)(5,103,11,125)(6,223,303,245)(7,224,304,241)(8,225,305,242)(9,221,301,243)(10,222,302,244)(16,233,313,266)(17,234,314,267)(18,235,315,268)(19,231,311,269)(20,232,312,270)(21,220,300,252)(22,216,296,253)(23,217,297,254)(24,218,298,255)(25,219,299,251)(26,131,51,99)(27,132,52,100)(28,133,53,96)(29,134,54,97)(30,135,55,98)(31,236,316,246)(32,237,317,247)(33,238,318,248)(34,239,319,249)(35,240,320,250)(36,126,46,116)(37,127,47,117)(38,128,48,118)(39,129,49,119)(40,130,50,120)(41,150,70,112)(42,146,66,113)(43,147,67,114)(44,148,68,115)(45,149,69,111)(56,173,93,136)(57,174,94,137)(58,175,95,138)(59,171,91,139)(60,172,92,140)(61,163,83,141)(62,164,84,142)(63,165,85,143)(64,161,81,144)(65,162,82,145)(71,189,109,151)(72,190,110,152)(73,186,106,153)(74,187,107,154)(75,188,108,155)(76,166,86,156)(77,167,87,157)(78,168,88,158)(79,169,89,159)(80,170,90,160)(176,256,213,293)(177,257,214,294)(178,258,215,295)(179,259,211,291)(180,260,212,292)(181,261,203,283)(182,262,204,284)(183,263,205,285)(184,264,201,281)(185,265,202,282)(191,271,229,309)(192,272,230,310)(193,273,226,306)(194,274,227,307)(195,275,228,308)(196,276,206,286)(197,277,207,287)(198,278,208,288)(199,279,209,289)(200,280,210,290), (1,184,12,201)(2,185,13,202)(3,181,14,203)(4,182,15,204)(5,183,11,205)(6,165,303,143)(7,161,304,144)(8,162,305,145)(9,163,301,141)(10,164,302,142)(16,186,313,153)(17,187,314,154)(18,188,315,155)(19,189,311,151)(20,190,312,152)(21,172,300,140)(22,173,296,136)(23,174,297,137)(24,175,298,138)(25,171,299,139)(26,211,51,179)(27,212,52,180)(28,213,53,176)(29,214,54,177)(30,215,55,178)(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)(41,230,70,192)(42,226,66,193)(43,227,67,194)(44,228,68,195)(45,229,69,191)(56,253,93,216)(57,254,94,217)(58,255,95,218)(59,251,91,219)(60,252,92,220)(61,243,83,221)(62,244,84,222)(63,245,85,223)(64,241,81,224)(65,242,82,225)(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)(96,293,133,256)(97,294,134,257)(98,295,135,258)(99,291,131,259)(100,292,132,260)(101,283,123,261)(102,284,124,262)(103,285,125,263)(104,281,121,264)(105,282,122,265)(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), (1,37,29,66)(2,38,30,67)(3,39,26,68)(4,40,27,69)(5,36,28,70)(6,316,296,20)(7,317,297,16)(8,318,298,17)(9,319,299,18)(10,320,300,19)(11,46,53,41)(12,47,54,42)(13,48,55,43)(14,49,51,44)(15,50,52,45)(21,311,302,35)(22,312,303,31)(23,313,304,32)(24,314,305,33)(25,315,301,34)(56,76,85,110)(57,77,81,106)(58,78,82,107)(59,79,83,108)(60,80,84,109)(61,75,91,89)(62,71,92,90)(63,72,93,86)(64,73,94,87)(65,74,95,88)(96,150,125,116)(97,146,121,117)(98,147,122,118)(99,148,123,119)(100,149,124,120)(101,129,131,115)(102,130,132,111)(103,126,133,112)(104,127,134,113)(105,128,135,114)(136,156,165,190)(137,157,161,186)(138,158,162,187)(139,159,163,188)(140,160,164,189)(141,155,171,169)(142,151,172,170)(143,152,173,166)(144,153,174,167)(145,154,175,168)(176,196,205,230)(177,197,201,226)(178,198,202,227)(179,199,203,228)(180,200,204,229)(181,195,211,209)(182,191,212,210)(183,192,213,206)(184,193,214,207)(185,194,215,208)(216,270,245,236)(217,266,241,237)(218,267,242,238)(219,268,243,239)(220,269,244,240)(221,249,251,235)(222,250,252,231)(223,246,253,232)(224,247,254,233)(225,248,255,234)(256,276,285,310)(257,277,281,306)(258,278,282,307)(259,279,283,308)(260,280,284,309)(261,275,291,289)(262,271,292,290)(263,272,293,286)(264,273,294,287)(265,274,295,288), (1,57,29,81)(2,58,30,82)(3,59,26,83)(4,60,27,84)(5,56,28,85)(6,276,296,310)(7,277,297,306)(8,278,298,307)(9,279,299,308)(10,280,300,309)(11,93,53,63)(12,94,54,64)(13,95,55,65)(14,91,51,61)(15,92,52,62)(16,281,317,257)(17,282,318,258)(18,283,319,259)(19,284,320,260)(20,285,316,256)(21,271,302,290)(22,272,303,286)(23,273,304,287)(24,274,305,288)(25,275,301,289)(31,293,312,263)(32,294,313,264)(33,295,314,265)(34,291,315,261)(35,292,311,262)(36,110,70,76)(37,106,66,77)(38,107,67,78)(39,108,68,79)(40,109,69,80)(41,86,46,72)(42,87,47,73)(43,88,48,74)(44,89,49,75)(45,90,50,71)(96,165,125,136)(97,161,121,137)(98,162,122,138)(99,163,123,139)(100,164,124,140)(101,171,131,141)(102,172,132,142)(103,173,133,143)(104,174,134,144)(105,175,135,145)(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)(176,270,205,236)(177,266,201,237)(178,267,202,238)(179,268,203,239)(180,269,204,240)(181,249,211,235)(182,250,212,231)(183,246,213,232)(184,247,214,233)(185,248,215,234)(191,222,210,252)(192,223,206,253)(193,224,207,254)(194,225,208,255)(195,221,209,251)(196,216,230,245)(197,217,226,241)(198,218,227,242)(199,219,228,243)(200,220,229,244), (1,104,12,121)(2,105,13,122)(3,101,14,123)(4,102,15,124)(5,103,11,125)(6,223,303,245)(7,224,304,241)(8,225,305,242)(9,221,301,243)(10,222,302,244)(16,233,313,266)(17,234,314,267)(18,235,315,268)(19,231,311,269)(20,232,312,270)(21,220,300,252)(22,216,296,253)(23,217,297,254)(24,218,298,255)(25,219,299,251)(26,131,51,99)(27,132,52,100)(28,133,53,96)(29,134,54,97)(30,135,55,98)(31,236,316,246)(32,237,317,247)(33,238,318,248)(34,239,319,249)(35,240,320,250)(36,126,46,116)(37,127,47,117)(38,128,48,118)(39,129,49,119)(40,130,50,120)(41,150,70,112)(42,146,66,113)(43,147,67,114)(44,148,68,115)(45,149,69,111)(56,173,93,136)(57,174,94,137)(58,175,95,138)(59,171,91,139)(60,172,92,140)(61,163,83,141)(62,164,84,142)(63,165,85,143)(64,161,81,144)(65,162,82,145)(71,189,109,151)(72,190,110,152)(73,186,106,153)(74,187,107,154)(75,188,108,155)(76,166,86,156)(77,167,87,157)(78,168,88,158)(79,169,89,159)(80,170,90,160)(176,256,213,293)(177,257,214,294)(178,258,215,295)(179,259,211,291)(180,260,212,292)(181,261,203,283)(182,262,204,284)(183,263,205,285)(184,264,201,281)(185,265,202,282)(191,271,229,309)(192,272,230,310)(193,273,226,306)(194,274,227,307)(195,275,228,308)(196,276,206,286)(197,277,207,287)(198,278,208,288)(199,279,209,289)(200,280,210,290), (1,184,12,201)(2,185,13,202)(3,181,14,203)(4,182,15,204)(5,183,11,205)(6,165,303,143)(7,161,304,144)(8,162,305,145)(9,163,301,141)(10,164,302,142)(16,186,313,153)(17,187,314,154)(18,188,315,155)(19,189,311,151)(20,190,312,152)(21,172,300,140)(22,173,296,136)(23,174,297,137)(24,175,298,138)(25,171,299,139)(26,211,51,179)(27,212,52,180)(28,213,53,176)(29,214,54,177)(30,215,55,178)(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)(41,230,70,192)(42,226,66,193)(43,227,67,194)(44,228,68,195)(45,229,69,191)(56,253,93,216)(57,254,94,217)(58,255,95,218)(59,251,91,219)(60,252,92,220)(61,243,83,221)(62,244,84,222)(63,245,85,223)(64,241,81,224)(65,242,82,225)(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)(96,293,133,256)(97,294,134,257)(98,295,135,258)(99,291,131,259)(100,292,132,260)(101,283,123,261)(102,284,124,262)(103,285,125,263)(104,281,121,264)(105,282,122,265)(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)], [(1,37,29,66),(2,38,30,67),(3,39,26,68),(4,40,27,69),(5,36,28,70),(6,316,296,20),(7,317,297,16),(8,318,298,17),(9,319,299,18),(10,320,300,19),(11,46,53,41),(12,47,54,42),(13,48,55,43),(14,49,51,44),(15,50,52,45),(21,311,302,35),(22,312,303,31),(23,313,304,32),(24,314,305,33),(25,315,301,34),(56,76,85,110),(57,77,81,106),(58,78,82,107),(59,79,83,108),(60,80,84,109),(61,75,91,89),(62,71,92,90),(63,72,93,86),(64,73,94,87),(65,74,95,88),(96,150,125,116),(97,146,121,117),(98,147,122,118),(99,148,123,119),(100,149,124,120),(101,129,131,115),(102,130,132,111),(103,126,133,112),(104,127,134,113),(105,128,135,114),(136,156,165,190),(137,157,161,186),(138,158,162,187),(139,159,163,188),(140,160,164,189),(141,155,171,169),(142,151,172,170),(143,152,173,166),(144,153,174,167),(145,154,175,168),(176,196,205,230),(177,197,201,226),(178,198,202,227),(179,199,203,228),(180,200,204,229),(181,195,211,209),(182,191,212,210),(183,192,213,206),(184,193,214,207),(185,194,215,208),(216,270,245,236),(217,266,241,237),(218,267,242,238),(219,268,243,239),(220,269,244,240),(221,249,251,235),(222,250,252,231),(223,246,253,232),(224,247,254,233),(225,248,255,234),(256,276,285,310),(257,277,281,306),(258,278,282,307),(259,279,283,308),(260,280,284,309),(261,275,291,289),(262,271,292,290),(263,272,293,286),(264,273,294,287),(265,274,295,288)], [(1,57,29,81),(2,58,30,82),(3,59,26,83),(4,60,27,84),(5,56,28,85),(6,276,296,310),(7,277,297,306),(8,278,298,307),(9,279,299,308),(10,280,300,309),(11,93,53,63),(12,94,54,64),(13,95,55,65),(14,91,51,61),(15,92,52,62),(16,281,317,257),(17,282,318,258),(18,283,319,259),(19,284,320,260),(20,285,316,256),(21,271,302,290),(22,272,303,286),(23,273,304,287),(24,274,305,288),(25,275,301,289),(31,293,312,263),(32,294,313,264),(33,295,314,265),(34,291,315,261),(35,292,311,262),(36,110,70,76),(37,106,66,77),(38,107,67,78),(39,108,68,79),(40,109,69,80),(41,86,46,72),(42,87,47,73),(43,88,48,74),(44,89,49,75),(45,90,50,71),(96,165,125,136),(97,161,121,137),(98,162,122,138),(99,163,123,139),(100,164,124,140),(101,171,131,141),(102,172,132,142),(103,173,133,143),(104,174,134,144),(105,175,135,145),(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),(176,270,205,236),(177,266,201,237),(178,267,202,238),(179,268,203,239),(180,269,204,240),(181,249,211,235),(182,250,212,231),(183,246,213,232),(184,247,214,233),(185,248,215,234),(191,222,210,252),(192,223,206,253),(193,224,207,254),(194,225,208,255),(195,221,209,251),(196,216,230,245),(197,217,226,241),(198,218,227,242),(199,219,228,243),(200,220,229,244)], [(1,104,12,121),(2,105,13,122),(3,101,14,123),(4,102,15,124),(5,103,11,125),(6,223,303,245),(7,224,304,241),(8,225,305,242),(9,221,301,243),(10,222,302,244),(16,233,313,266),(17,234,314,267),(18,235,315,268),(19,231,311,269),(20,232,312,270),(21,220,300,252),(22,216,296,253),(23,217,297,254),(24,218,298,255),(25,219,299,251),(26,131,51,99),(27,132,52,100),(28,133,53,96),(29,134,54,97),(30,135,55,98),(31,236,316,246),(32,237,317,247),(33,238,318,248),(34,239,319,249),(35,240,320,250),(36,126,46,116),(37,127,47,117),(38,128,48,118),(39,129,49,119),(40,130,50,120),(41,150,70,112),(42,146,66,113),(43,147,67,114),(44,148,68,115),(45,149,69,111),(56,173,93,136),(57,174,94,137),(58,175,95,138),(59,171,91,139),(60,172,92,140),(61,163,83,141),(62,164,84,142),(63,165,85,143),(64,161,81,144),(65,162,82,145),(71,189,109,151),(72,190,110,152),(73,186,106,153),(74,187,107,154),(75,188,108,155),(76,166,86,156),(77,167,87,157),(78,168,88,158),(79,169,89,159),(80,170,90,160),(176,256,213,293),(177,257,214,294),(178,258,215,295),(179,259,211,291),(180,260,212,292),(181,261,203,283),(182,262,204,284),(183,263,205,285),(184,264,201,281),(185,265,202,282),(191,271,229,309),(192,272,230,310),(193,273,226,306),(194,274,227,307),(195,275,228,308),(196,276,206,286),(197,277,207,287),(198,278,208,288),(199,279,209,289),(200,280,210,290)], [(1,184,12,201),(2,185,13,202),(3,181,14,203),(4,182,15,204),(5,183,11,205),(6,165,303,143),(7,161,304,144),(8,162,305,145),(9,163,301,141),(10,164,302,142),(16,186,313,153),(17,187,314,154),(18,188,315,155),(19,189,311,151),(20,190,312,152),(21,172,300,140),(22,173,296,136),(23,174,297,137),(24,175,298,138),(25,171,299,139),(26,211,51,179),(27,212,52,180),(28,213,53,176),(29,214,54,177),(30,215,55,178),(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),(41,230,70,192),(42,226,66,193),(43,227,67,194),(44,228,68,195),(45,229,69,191),(56,253,93,216),(57,254,94,217),(58,255,95,218),(59,251,91,219),(60,252,92,220),(61,243,83,221),(62,244,84,222),(63,245,85,223),(64,241,81,224),(65,242,82,225),(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),(96,293,133,256),(97,294,134,257),(98,295,135,258),(99,291,131,259),(100,292,132,260),(101,283,123,261),(102,284,124,262),(103,285,125,263),(104,281,121,264),(105,282,122,265),(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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | Q8 | SD16 | C4○D4 | C5×D4 | C5×Q8 | C5×SD16 | C5×C4○D4 | C8.C22 | C5×C8.C22 |
| kernel | C5×Q8⋊Q8 | C5×Q8⋊C4 | C5×C4⋊C8 | C5×C4.Q8 | Q8×C20 | C5×C4⋊Q8 | Q8⋊Q8 | Q8⋊C4 | C4⋊C8 | C4.Q8 | C4×Q8 | C4⋊Q8 | C2×C20 | C5×Q8 | C20 | C20 | C2×C4 | Q8 | C4 | C4 | C10 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 8 | 4 | 8 | 4 | 4 | 2 | 2 | 4 | 2 | 8 | 8 | 16 | 8 | 1 | 4 |
Matrix representation of C5×Q8⋊Q8 ►in GL4(𝔽41) generated by
| 37 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 14 | 34 |
| 0 | 0 | 34 | 27 |
| 1 | 39 | 0 | 0 |
| 1 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 23 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 23 | 13 |
| 0 | 0 | 13 | 18 |
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,40,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,14,34,0,0,34,27],[1,1,0,0,39,40,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,23,32,0,0,0,0,23,13,0,0,13,18] >;
C5×Q8⋊Q8 in GAP, Magma, Sage, TeX
C_5\times Q_8\rtimes Q_8
% in TeX
G:=Group("C5xQ8:Q8"); // GroupNames label
G:=SmallGroup(320,976);
// by ID
G=gap.SmallGroup(320,976);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,280,589,288,1766,856,10085,2539,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=d^4=1,c^2=b^2,e^2=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=d^-1>;
// generators/relations