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G = C2×C52⋊4Q8  order 400 = 24·52

Direct product of C2 and C52⋊4Q8

direct product, metabelian, supersoluble, monomial

Aliases: C2×C52⋊4Q8, C10⋊2Dic10, C20.49D10, C102.31C22, (C5×C10)⋊4Q8, C52⋊6(C2×Q8), (C2×C20).9D5, (C10×C20).5C2, C5⋊3(C2×Dic10), (C2×C10).35D10, (C5×C20).35C22, (C5×C10).29C23, C10.30(C22×D5), C52⋊6C4.14C22, C4.11(C2×C5⋊D5), (C2×C4).4(C5⋊D5), C2.3(C22×C5⋊D5), C22.8(C2×C5⋊D5), (C2×C52⋊6C4).7C2, SmallGroup(400,191)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5×C10 — C2×C52⋊4Q8
Chief series
C1 — C5 — C52 — C5×C10 — C52⋊6C4 — C2×C52⋊6C4 — C2×C52⋊4Q8
Lower central
C52 — C5×C10 — C2×C52⋊4Q8
Upper central
C1 — C22 — C2×C4

Generators and relations for C2×C52⋊4Q8
 G = < a,b,c,d,e | a2=b5=c5=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b-1, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 680 in 152 conjugacy classes, 75 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C5, C2×C4, C2×C4, Q8, C10, C2×Q8, Dic5, C20, C2×C10, C52, Dic10, C2×Dic5, C2×C20, C5×C10, C5×C10, C2×Dic10, C52⋊6C4, C5×C20, C102, C52⋊4Q8, C2×C52⋊6C4, C10×C20, C2×C52⋊4Q8
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, D10, Dic10, C22×D5, C5⋊D5, C2×Dic10, C2×C5⋊D5, C52⋊4Q8, C22×C5⋊D5, C2×C52⋊4Q8

Smallest permutation representation of C2×C52⋊4Q8
►Regular action on 400 points
Generators in S400
(1 148)(2 149)(3 150)(4 146)(5 147)(6 295)(7 291)(8 292)(9 293)(10 294)(11 290)(12 286)(13 287)(14 288)(15 289)(16 285)(17 281)(18 282)(19 283)(20 284)(21 280)(22 276)(23 277)(24 278)(25 279)(26 151)(27 152)(28 153)(29 154)(30 155)(31 156)(32 157)(33 158)(34 159)(35 160)(36 161)(37 162)(38 163)(39 164)(40 165)(41 166)(42 167)(43 168)(44 169)(45 170)(46 121)(47 122)(48 123)(49 124)(50 125)(51 126)(52 127)(53 128)(54 129)(55 130)(56 131)(57 132)(58 133)(59 134)(60 135)(61 136)(62 137)(63 138)(64 139)(65 140)(66 141)(67 142)(68 143)(69 144)(70 145)(71 196)(72 197)(73 198)(74 199)(75 200)(76 216)(77 217)(78 218)(79 219)(80 220)(81 211)(82 212)(83 213)(84 214)(85 215)(86 206)(87 207)(88 208)(89 209)(90 210)(91 201)(92 202)(93 203)(94 204)(95 205)(96 171)(97 172)(98 173)(99 174)(100 175)(101 176)(102 177)(103 178)(104 179)(105 180)(106 181)(107 182)(108 183)(109 184)(110 185)(111 186)(112 187)(113 188)(114 189)(115 190)(116 191)(117 192)(118 193)(119 194)(120 195)(221 346)(222 347)(223 348)(224 349)(225 350)(226 351)(227 352)(228 353)(229 354)(230 355)(231 356)(232 357)(233 358)(234 359)(235 360)(236 361)(237 362)(238 363)(239 364)(240 365)(241 366)(242 367)(243 368)(244 369)(245 370)(246 321)(247 322)(248 323)(249 324)(250 325)(251 326)(252 327)(253 328)(254 329)(255 330)(256 331)(257 332)(258 333)(259 334)(260 335)(261 336)(262 337)(263 338)(264 339)(265 340)(266 341)(267 342)(268 343)(269 344)(270 345)(271 396)(272 397)(273 398)(274 399)(275 400)(296 371)(297 372)(298 373)(299 374)(300 375)(301 376)(302 377)(303 378)(304 379)(305 380)(306 381)(307 382)(308 383)(309 384)(310 385)(311 386)(312 387)(313 388)(314 389)(315 390)(316 391)(317 392)(318 393)(319 394)(320 395)

(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)(321 322 323 324 325)(326 327 328 329 330)(331 332 333 334 335)(336 337 338 339 340)(341 342 343 344 345)(346 347 348 349 350)(351 352 353 354 355)(356 357 358 359 360)(361 362 363 364 365)(366 367 368 369 370)(371 372 373 374 375)(376 377 378 379 380)(381 382 383 384 385)(386 387 388 389 390)(391 392 393 394 395)(396 397 398 399 400)

(1 44 33 28 38)(2 45 34 29 39)(3 41 35 30 40)(4 42 31 26 36)(5 43 32 27 37)(6 16 400 25 11)(7 17 396 21 12)(8 18 397 22 13)(9 19 398 23 14)(10 20 399 24 15)(46 67 56 51 61)(47 68 57 52 62)(48 69 58 53 63)(49 70 59 54 64)(50 66 60 55 65)(71 92 81 76 86)(72 93 82 77 87)(73 94 83 78 88)(74 95 84 79 89)(75 91 85 80 90)(96 117 106 101 111)(97 118 107 102 112)(98 119 108 103 113)(99 120 109 104 114)(100 116 110 105 115)(121 142 131 126 136)(122 143 132 127 137)(123 144 133 128 138)(124 145 134 129 139)(125 141 135 130 140)(146 167 156 151 161)(147 168 157 152 162)(148 169 158 153 163)(149 170 159 154 164)(150 166 160 155 165)(171 192 181 176 186)(172 193 182 177 187)(173 194 183 178 188)(174 195 184 179 189)(175 191 185 180 190)(196 202 211 216 206)(197 203 212 217 207)(198 204 213 218 208)(199 205 214 219 209)(200 201 215 220 210)(221 230 236 241 231)(222 226 237 242 232)(223 227 238 243 233)(224 228 239 244 234)(225 229 240 245 235)(246 255 261 266 256)(247 251 262 267 257)(248 252 263 268 258)(249 253 264 269 259)(250 254 265 270 260)(271 280 286 291 281)(272 276 287 292 282)(273 277 288 293 283)(274 278 289 294 284)(275 279 290 295 285)(296 305 311 316 306)(297 301 312 317 307)(298 302 313 318 308)(299 303 314 319 309)(300 304 315 320 310)(321 330 336 341 331)(322 326 337 342 332)(323 327 338 343 333)(324 328 339 344 334)(325 329 340 345 335)(346 355 361 366 356)(347 351 362 367 357)(348 352 363 368 358)(349 353 364 369 359)(350 354 365 370 360)(371 380 386 391 381)(372 376 387 392 382)(373 377 388 393 383)(374 378 389 394 384)(375 379 390 395 385)

(1 98 48 73)(2 99 49 74)(3 100 50 75)(4 96 46 71)(5 97 47 72)(6 345 395 370)(7 341 391 366)(8 342 392 367)(9 343 393 368)(10 344 394 369)(11 340 390 365)(12 336 386 361)(13 337 387 362)(14 338 388 363)(15 339 389 364)(16 335 385 360)(17 331 381 356)(18 332 382 357)(19 333 383 358)(20 334 384 359)(21 330 380 355)(22 326 376 351)(23 327 377 352)(24 328 378 353)(25 329 379 354)(26 101 51 76)(27 102 52 77)(28 103 53 78)(29 104 54 79)(30 105 55 80)(31 106 56 81)(32 107 57 82)(33 108 58 83)(34 109 59 84)(35 110 60 85)(36 111 61 86)(37 112 62 87)(38 113 63 88)(39 114 64 89)(40 115 65 90)(41 116 66 91)(42 117 67 92)(43 118 68 93)(44 119 69 94)(45 120 70 95)(121 196 146 171)(122 197 147 172)(123 198 148 173)(124 199 149 174)(125 200 150 175)(126 216 151 176)(127 217 152 177)(128 218 153 178)(129 219 154 179)(130 220 155 180)(131 211 156 181)(132 212 157 182)(133 213 158 183)(134 214 159 184)(135 215 160 185)(136 206 161 186)(137 207 162 187)(138 208 163 188)(139 209 164 189)(140 210 165 190)(141 201 166 191)(142 202 167 192)(143 203 168 193)(144 204 169 194)(145 205 170 195)(221 271 246 296)(222 272 247 297)(223 273 248 298)(224 274 249 299)(225 275 250 300)(226 276 251 301)(227 277 252 302)(228 278 253 303)(229 279 254 304)(230 280 255 305)(231 281 256 306)(232 282 257 307)(233 283 258 308)(234 284 259 309)(235 285 260 310)(236 286 261 311)(237 287 262 312)(238 288 263 313)(239 289 264 314)(240 290 265 315)(241 291 266 316)(242 292 267 317)(243 293 268 318)(244 294 269 319)(245 295 270 320)(321 371 346 396)(322 372 347 397)(323 373 348 398)(324 374 349 399)(325 375 350 400)

(1 262 48 237)(2 261 49 236)(3 265 50 240)(4 264 46 239)(5 263 47 238)(6 210 395 190)(7 209 391 189)(8 208 392 188)(9 207 393 187)(10 206 394 186)(11 200 390 175)(12 199 386 174)(13 198 387 173)(14 197 388 172)(15 196 389 171)(16 220 385 180)(17 219 381 179)(18 218 382 178)(19 217 383 177)(20 216 384 176)(21 205 380 195)(22 204 376 194)(23 203 377 193)(24 202 378 192)(25 201 379 191)(26 259 51 234)(27 258 52 233)(28 257 53 232)(29 256 54 231)(30 260 55 235)(31 249 56 224)(32 248 57 223)(33 247 58 222)(34 246 59 221)(35 250 60 225)(36 269 61 244)(37 268 62 243)(38 267 63 242)(39 266 64 241)(40 270 65 245)(41 254 66 229)(42 253 67 228)(43 252 68 227)(44 251 69 226)(45 255 70 230)(71 314 96 289)(72 313 97 288)(73 312 98 287)(74 311 99 286)(75 315 100 290)(76 309 101 284)(77 308 102 283)(78 307 103 282)(79 306 104 281)(80 310 105 285)(81 299 106 274)(82 298 107 273)(83 297 108 272)(84 296 109 271)(85 300 110 275)(86 319 111 294)(87 318 112 293)(88 317 113 292)(89 316 114 291)(90 320 115 295)(91 304 116 279)(92 303 117 278)(93 302 118 277)(94 301 119 276)(95 305 120 280)(121 364 146 339)(122 363 147 338)(123 362 148 337)(124 361 149 336)(125 365 150 340)(126 359 151 334)(127 358 152 333)(128 357 153 332)(129 356 154 331)(130 360 155 335)(131 349 156 324)(132 348 157 323)(133 347 158 322)(134 346 159 321)(135 350 160 325)(136 369 161 344)(137 368 162 343)(138 367 163 342)(139 366 164 341)(140 370 165 345)(141 354 166 329)(142 353 167 328)(143 352 168 327)(144 351 169 326)(145 355 170 330)(181 399 211 374)(182 398 212 373)(183 397 213 372)(184 396 214 371)(185 400 215 375)
 
G:=sub<Sym(400)| (1,148)(2,149)(3,150)(4,146)(5,147)(6,295)(7,291)(8,292)(9,293)(10,294)(11,290)(12,286)(13,287)(14,288)(15,289)(16,285)(17,281)(18,282)(19,283)(20,284)(21,280)(22,276)(23,277)(24,278)(25,279)(26,151)(27,152)(28,153)(29,154)(30,155)(31,156)(32,157)(33,158)(34,159)(35,160)(36,161)(37,162)(38,163)(39,164)(40,165)(41,166)(42,167)(43,168)(44,169)(45,170)(46,121)(47,122)(48,123)(49,124)(50,125)(51,126)(52,127)(53,128)(54,129)(55,130)(56,131)(57,132)(58,133)(59,134)(60,135)(61,136)(62,137)(63,138)(64,139)(65,140)(66,141)(67,142)(68,143)(69,144)(70,145)(71,196)(72,197)(73,198)(74,199)(75,200)(76,216)(77,217)(78,218)(79,219)(80,220)(81,211)(82,212)(83,213)(84,214)(85,215)(86,206)(87,207)(88,208)(89,209)(90,210)(91,201)(92,202)(93,203)(94,204)(95,205)(96,171)(97,172)(98,173)(99,174)(100,175)(101,176)(102,177)(103,178)(104,179)(105,180)(106,181)(107,182)(108,183)(109,184)(110,185)(111,186)(112,187)(113,188)(114,189)(115,190)(116,191)(117,192)(118,193)(119,194)(120,195)(221,346)(222,347)(223,348)(224,349)(225,350)(226,351)(227,352)(228,353)(229,354)(230,355)(231,356)(232,357)(233,358)(234,359)(235,360)(236,361)(237,362)(238,363)(239,364)(240,365)(241,366)(242,367)(243,368)(244,369)(245,370)(246,321)(247,322)(248,323)(249,324)(250,325)(251,326)(252,327)(253,328)(254,329)(255,330)(256,331)(257,332)(258,333)(259,334)(260,335)(261,336)(262,337)(263,338)(264,339)(265,340)(266,341)(267,342)(268,343)(269,344)(270,345)(271,396)(272,397)(273,398)(274,399)(275,400)(296,371)(297,372)(298,373)(299,374)(300,375)(301,376)(302,377)(303,378)(304,379)(305,380)(306,381)(307,382)(308,383)(309,384)(310,385)(311,386)(312,387)(313,388)(314,389)(315,390)(316,391)(317,392)(318,393)(319,394)(320,395), (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)(321,322,323,324,325)(326,327,328,329,330)(331,332,333,334,335)(336,337,338,339,340)(341,342,343,344,345)(346,347,348,349,350)(351,352,353,354,355)(356,357,358,359,360)(361,362,363,364,365)(366,367,368,369,370)(371,372,373,374,375)(376,377,378,379,380)(381,382,383,384,385)(386,387,388,389,390)(391,392,393,394,395)(396,397,398,399,400), (1,44,33,28,38)(2,45,34,29,39)(3,41,35,30,40)(4,42,31,26,36)(5,43,32,27,37)(6,16,400,25,11)(7,17,396,21,12)(8,18,397,22,13)(9,19,398,23,14)(10,20,399,24,15)(46,67,56,51,61)(47,68,57,52,62)(48,69,58,53,63)(49,70,59,54,64)(50,66,60,55,65)(71,92,81,76,86)(72,93,82,77,87)(73,94,83,78,88)(74,95,84,79,89)(75,91,85,80,90)(96,117,106,101,111)(97,118,107,102,112)(98,119,108,103,113)(99,120,109,104,114)(100,116,110,105,115)(121,142,131,126,136)(122,143,132,127,137)(123,144,133,128,138)(124,145,134,129,139)(125,141,135,130,140)(146,167,156,151,161)(147,168,157,152,162)(148,169,158,153,163)(149,170,159,154,164)(150,166,160,155,165)(171,192,181,176,186)(172,193,182,177,187)(173,194,183,178,188)(174,195,184,179,189)(175,191,185,180,190)(196,202,211,216,206)(197,203,212,217,207)(198,204,213,218,208)(199,205,214,219,209)(200,201,215,220,210)(221,230,236,241,231)(222,226,237,242,232)(223,227,238,243,233)(224,228,239,244,234)(225,229,240,245,235)(246,255,261,266,256)(247,251,262,267,257)(248,252,263,268,258)(249,253,264,269,259)(250,254,265,270,260)(271,280,286,291,281)(272,276,287,292,282)(273,277,288,293,283)(274,278,289,294,284)(275,279,290,295,285)(296,305,311,316,306)(297,301,312,317,307)(298,302,313,318,308)(299,303,314,319,309)(300,304,315,320,310)(321,330,336,341,331)(322,326,337,342,332)(323,327,338,343,333)(324,328,339,344,334)(325,329,340,345,335)(346,355,361,366,356)(347,351,362,367,357)(348,352,363,368,358)(349,353,364,369,359)(350,354,365,370,360)(371,380,386,391,381)(372,376,387,392,382)(373,377,388,393,383)(374,378,389,394,384)(375,379,390,395,385), (1,98,48,73)(2,99,49,74)(3,100,50,75)(4,96,46,71)(5,97,47,72)(6,345,395,370)(7,341,391,366)(8,342,392,367)(9,343,393,368)(10,344,394,369)(11,340,390,365)(12,336,386,361)(13,337,387,362)(14,338,388,363)(15,339,389,364)(16,335,385,360)(17,331,381,356)(18,332,382,357)(19,333,383,358)(20,334,384,359)(21,330,380,355)(22,326,376,351)(23,327,377,352)(24,328,378,353)(25,329,379,354)(26,101,51,76)(27,102,52,77)(28,103,53,78)(29,104,54,79)(30,105,55,80)(31,106,56,81)(32,107,57,82)(33,108,58,83)(34,109,59,84)(35,110,60,85)(36,111,61,86)(37,112,62,87)(38,113,63,88)(39,114,64,89)(40,115,65,90)(41,116,66,91)(42,117,67,92)(43,118,68,93)(44,119,69,94)(45,120,70,95)(121,196,146,171)(122,197,147,172)(123,198,148,173)(124,199,149,174)(125,200,150,175)(126,216,151,176)(127,217,152,177)(128,218,153,178)(129,219,154,179)(130,220,155,180)(131,211,156,181)(132,212,157,182)(133,213,158,183)(134,214,159,184)(135,215,160,185)(136,206,161,186)(137,207,162,187)(138,208,163,188)(139,209,164,189)(140,210,165,190)(141,201,166,191)(142,202,167,192)(143,203,168,193)(144,204,169,194)(145,205,170,195)(221,271,246,296)(222,272,247,297)(223,273,248,298)(224,274,249,299)(225,275,250,300)(226,276,251,301)(227,277,252,302)(228,278,253,303)(229,279,254,304)(230,280,255,305)(231,281,256,306)(232,282,257,307)(233,283,258,308)(234,284,259,309)(235,285,260,310)(236,286,261,311)(237,287,262,312)(238,288,263,313)(239,289,264,314)(240,290,265,315)(241,291,266,316)(242,292,267,317)(243,293,268,318)(244,294,269,319)(245,295,270,320)(321,371,346,396)(322,372,347,397)(323,373,348,398)(324,374,349,399)(325,375,350,400), (1,262,48,237)(2,261,49,236)(3,265,50,240)(4,264,46,239)(5,263,47,238)(6,210,395,190)(7,209,391,189)(8,208,392,188)(9,207,393,187)(10,206,394,186)(11,200,390,175)(12,199,386,174)(13,198,387,173)(14,197,388,172)(15,196,389,171)(16,220,385,180)(17,219,381,179)(18,218,382,178)(19,217,383,177)(20,216,384,176)(21,205,380,195)(22,204,376,194)(23,203,377,193)(24,202,378,192)(25,201,379,191)(26,259,51,234)(27,258,52,233)(28,257,53,232)(29,256,54,231)(30,260,55,235)(31,249,56,224)(32,248,57,223)(33,247,58,222)(34,246,59,221)(35,250,60,225)(36,269,61,244)(37,268,62,243)(38,267,63,242)(39,266,64,241)(40,270,65,245)(41,254,66,229)(42,253,67,228)(43,252,68,227)(44,251,69,226)(45,255,70,230)(71,314,96,289)(72,313,97,288)(73,312,98,287)(74,311,99,286)(75,315,100,290)(76,309,101,284)(77,308,102,283)(78,307,103,282)(79,306,104,281)(80,310,105,285)(81,299,106,274)(82,298,107,273)(83,297,108,272)(84,296,109,271)(85,300,110,275)(86,319,111,294)(87,318,112,293)(88,317,113,292)(89,316,114,291)(90,320,115,295)(91,304,116,279)(92,303,117,278)(93,302,118,277)(94,301,119,276)(95,305,120,280)(121,364,146,339)(122,363,147,338)(123,362,148,337)(124,361,149,336)(125,365,150,340)(126,359,151,334)(127,358,152,333)(128,357,153,332)(129,356,154,331)(130,360,155,335)(131,349,156,324)(132,348,157,323)(133,347,158,322)(134,346,159,321)(135,350,160,325)(136,369,161,344)(137,368,162,343)(138,367,163,342)(139,366,164,341)(140,370,165,345)(141,354,166,329)(142,353,167,328)(143,352,168,327)(144,351,169,326)(145,355,170,330)(181,399,211,374)(182,398,212,373)(183,397,213,372)(184,396,214,371)(185,400,215,375)>;
 
G:=Group( (1,148)(2,149)(3,150)(4,146)(5,147)(6,295)(7,291)(8,292)(9,293)(10,294)(11,290)(12,286)(13,287)(14,288)(15,289)(16,285)(17,281)(18,282)(19,283)(20,284)(21,280)(22,276)(23,277)(24,278)(25,279)(26,151)(27,152)(28,153)(29,154)(30,155)(31,156)(32,157)(33,158)(34,159)(35,160)(36,161)(37,162)(38,163)(39,164)(40,165)(41,166)(42,167)(43,168)(44,169)(45,170)(46,121)(47,122)(48,123)(49,124)(50,125)(51,126)(52,127)(53,128)(54,129)(55,130)(56,131)(57,132)(58,133)(59,134)(60,135)(61,136)(62,137)(63,138)(64,139)(65,140)(66,141)(67,142)(68,143)(69,144)(70,145)(71,196)(72,197)(73,198)(74,199)(75,200)(76,216)(77,217)(78,218)(79,219)(80,220)(81,211)(82,212)(83,213)(84,214)(85,215)(86,206)(87,207)(88,208)(89,209)(90,210)(91,201)(92,202)(93,203)(94,204)(95,205)(96,171)(97,172)(98,173)(99,174)(100,175)(101,176)(102,177)(103,178)(104,179)(105,180)(106,181)(107,182)(108,183)(109,184)(110,185)(111,186)(112,187)(113,188)(114,189)(115,190)(116,191)(117,192)(118,193)(119,194)(120,195)(221,346)(222,347)(223,348)(224,349)(225,350)(226,351)(227,352)(228,353)(229,354)(230,355)(231,356)(232,357)(233,358)(234,359)(235,360)(236,361)(237,362)(238,363)(239,364)(240,365)(241,366)(242,367)(243,368)(244,369)(245,370)(246,321)(247,322)(248,323)(249,324)(250,325)(251,326)(252,327)(253,328)(254,329)(255,330)(256,331)(257,332)(258,333)(259,334)(260,335)(261,336)(262,337)(263,338)(264,339)(265,340)(266,341)(267,342)(268,343)(269,344)(270,345)(271,396)(272,397)(273,398)(274,399)(275,400)(296,371)(297,372)(298,373)(299,374)(300,375)(301,376)(302,377)(303,378)(304,379)(305,380)(306,381)(307,382)(308,383)(309,384)(310,385)(311,386)(312,387)(313,388)(314,389)(315,390)(316,391)(317,392)(318,393)(319,394)(320,395), (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)(321,322,323,324,325)(326,327,328,329,330)(331,332,333,334,335)(336,337,338,339,340)(341,342,343,344,345)(346,347,348,349,350)(351,352,353,354,355)(356,357,358,359,360)(361,362,363,364,365)(366,367,368,369,370)(371,372,373,374,375)(376,377,378,379,380)(381,382,383,384,385)(386,387,388,389,390)(391,392,393,394,395)(396,397,398,399,400), (1,44,33,28,38)(2,45,34,29,39)(3,41,35,30,40)(4,42,31,26,36)(5,43,32,27,37)(6,16,400,25,11)(7,17,396,21,12)(8,18,397,22,13)(9,19,398,23,14)(10,20,399,24,15)(46,67,56,51,61)(47,68,57,52,62)(48,69,58,53,63)(49,70,59,54,64)(50,66,60,55,65)(71,92,81,76,86)(72,93,82,77,87)(73,94,83,78,88)(74,95,84,79,89)(75,91,85,80,90)(96,117,106,101,111)(97,118,107,102,112)(98,119,108,103,113)(99,120,109,104,114)(100,116,110,105,115)(121,142,131,126,136)(122,143,132,127,137)(123,144,133,128,138)(124,145,134,129,139)(125,141,135,130,140)(146,167,156,151,161)(147,168,157,152,162)(148,169,158,153,163)(149,170,159,154,164)(150,166,160,155,165)(171,192,181,176,186)(172,193,182,177,187)(173,194,183,178,188)(174,195,184,179,189)(175,191,185,180,190)(196,202,211,216,206)(197,203,212,217,207)(198,204,213,218,208)(199,205,214,219,209)(200,201,215,220,210)(221,230,236,241,231)(222,226,237,242,232)(223,227,238,243,233)(224,228,239,244,234)(225,229,240,245,235)(246,255,261,266,256)(247,251,262,267,257)(248,252,263,268,258)(249,253,264,269,259)(250,254,265,270,260)(271,280,286,291,281)(272,276,287,292,282)(273,277,288,293,283)(274,278,289,294,284)(275,279,290,295,285)(296,305,311,316,306)(297,301,312,317,307)(298,302,313,318,308)(299,303,314,319,309)(300,304,315,320,310)(321,330,336,341,331)(322,326,337,342,332)(323,327,338,343,333)(324,328,339,344,334)(325,329,340,345,335)(346,355,361,366,356)(347,351,362,367,357)(348,352,363,368,358)(349,353,364,369,359)(350,354,365,370,360)(371,380,386,391,381)(372,376,387,392,382)(373,377,388,393,383)(374,378,389,394,384)(375,379,390,395,385), (1,98,48,73)(2,99,49,74)(3,100,50,75)(4,96,46,71)(5,97,47,72)(6,345,395,370)(7,341,391,366)(8,342,392,367)(9,343,393,368)(10,344,394,369)(11,340,390,365)(12,336,386,361)(13,337,387,362)(14,338,388,363)(15,339,389,364)(16,335,385,360)(17,331,381,356)(18,332,382,357)(19,333,383,358)(20,334,384,359)(21,330,380,355)(22,326,376,351)(23,327,377,352)(24,328,378,353)(25,329,379,354)(26,101,51,76)(27,102,52,77)(28,103,53,78)(29,104,54,79)(30,105,55,80)(31,106,56,81)(32,107,57,82)(33,108,58,83)(34,109,59,84)(35,110,60,85)(36,111,61,86)(37,112,62,87)(38,113,63,88)(39,114,64,89)(40,115,65,90)(41,116,66,91)(42,117,67,92)(43,118,68,93)(44,119,69,94)(45,120,70,95)(121,196,146,171)(122,197,147,172)(123,198,148,173)(124,199,149,174)(125,200,150,175)(126,216,151,176)(127,217,152,177)(128,218,153,178)(129,219,154,179)(130,220,155,180)(131,211,156,181)(132,212,157,182)(133,213,158,183)(134,214,159,184)(135,215,160,185)(136,206,161,186)(137,207,162,187)(138,208,163,188)(139,209,164,189)(140,210,165,190)(141,201,166,191)(142,202,167,192)(143,203,168,193)(144,204,169,194)(145,205,170,195)(221,271,246,296)(222,272,247,297)(223,273,248,298)(224,274,249,299)(225,275,250,300)(226,276,251,301)(227,277,252,302)(228,278,253,303)(229,279,254,304)(230,280,255,305)(231,281,256,306)(232,282,257,307)(233,283,258,308)(234,284,259,309)(235,285,260,310)(236,286,261,311)(237,287,262,312)(238,288,263,313)(239,289,264,314)(240,290,265,315)(241,291,266,316)(242,292,267,317)(243,293,268,318)(244,294,269,319)(245,295,270,320)(321,371,346,396)(322,372,347,397)(323,373,348,398)(324,374,349,399)(325,375,350,400), (1,262,48,237)(2,261,49,236)(3,265,50,240)(4,264,46,239)(5,263,47,238)(6,210,395,190)(7,209,391,189)(8,208,392,188)(9,207,393,187)(10,206,394,186)(11,200,390,175)(12,199,386,174)(13,198,387,173)(14,197,388,172)(15,196,389,171)(16,220,385,180)(17,219,381,179)(18,218,382,178)(19,217,383,177)(20,216,384,176)(21,205,380,195)(22,204,376,194)(23,203,377,193)(24,202,378,192)(25,201,379,191)(26,259,51,234)(27,258,52,233)(28,257,53,232)(29,256,54,231)(30,260,55,235)(31,249,56,224)(32,248,57,223)(33,247,58,222)(34,246,59,221)(35,250,60,225)(36,269,61,244)(37,268,62,243)(38,267,63,242)(39,266,64,241)(40,270,65,245)(41,254,66,229)(42,253,67,228)(43,252,68,227)(44,251,69,226)(45,255,70,230)(71,314,96,289)(72,313,97,288)(73,312,98,287)(74,311,99,286)(75,315,100,290)(76,309,101,284)(77,308,102,283)(78,307,103,282)(79,306,104,281)(80,310,105,285)(81,299,106,274)(82,298,107,273)(83,297,108,272)(84,296,109,271)(85,300,110,275)(86,319,111,294)(87,318,112,293)(88,317,113,292)(89,316,114,291)(90,320,115,295)(91,304,116,279)(92,303,117,278)(93,302,118,277)(94,301,119,276)(95,305,120,280)(121,364,146,339)(122,363,147,338)(123,362,148,337)(124,361,149,336)(125,365,150,340)(126,359,151,334)(127,358,152,333)(128,357,153,332)(129,356,154,331)(130,360,155,335)(131,349,156,324)(132,348,157,323)(133,347,158,322)(134,346,159,321)(135,350,160,325)(136,369,161,344)(137,368,162,343)(138,367,163,342)(139,366,164,341)(140,370,165,345)(141,354,166,329)(142,353,167,328)(143,352,168,327)(144,351,169,326)(145,355,170,330)(181,399,211,374)(182,398,212,373)(183,397,213,372)(184,396,214,371)(185,400,215,375) );
 
G=PermutationGroup([[(1,148),(2,149),(3,150),(4,146),(5,147),(6,295),(7,291),(8,292),(9,293),(10,294),(11,290),(12,286),(13,287),(14,288),(15,289),(16,285),(17,281),(18,282),(19,283),(20,284),(21,280),(22,276),(23,277),(24,278),(25,279),(26,151),(27,152),(28,153),(29,154),(30,155),(31,156),(32,157),(33,158),(34,159),(35,160),(36,161),(37,162),(38,163),(39,164),(40,165),(41,166),(42,167),(43,168),(44,169),(45,170),(46,121),(47,122),(48,123),(49,124),(50,125),(51,126),(52,127),(53,128),(54,129),(55,130),(56,131),(57,132),(58,133),(59,134),(60,135),(61,136),(62,137),(63,138),(64,139),(65,140),(66,141),(67,142),(68,143),(69,144),(70,145),(71,196),(72,197),(73,198),(74,199),(75,200),(76,216),(77,217),(78,218),(79,219),(80,220),(81,211),(82,212),(83,213),(84,214),(85,215),(86,206),(87,207),(88,208),(89,209),(90,210),(91,201),(92,202),(93,203),(94,204),(95,205),(96,171),(97,172),(98,173),(99,174),(100,175),(101,176),(102,177),(103,178),(104,179),(105,180),(106,181),(107,182),(108,183),(109,184),(110,185),(111,186),(112,187),(113,188),(114,189),(115,190),(116,191),(117,192),(118,193),(119,194),(120,195),(221,346),(222,347),(223,348),(224,349),(225,350),(226,351),(227,352),(228,353),(229,354),(230,355),(231,356),(232,357),(233,358),(234,359),(235,360),(236,361),(237,362),(238,363),(239,364),(240,365),(241,366),(242,367),(243,368),(244,369),(245,370),(246,321),(247,322),(248,323),(249,324),(250,325),(251,326),(252,327),(253,328),(254,329),(255,330),(256,331),(257,332),(258,333),(259,334),(260,335),(261,336),(262,337),(263,338),(264,339),(265,340),(266,341),(267,342),(268,343),(269,344),(270,345),(271,396),(272,397),(273,398),(274,399),(275,400),(296,371),(297,372),(298,373),(299,374),(300,375),(301,376),(302,377),(303,378),(304,379),(305,380),(306,381),(307,382),(308,383),(309,384),(310,385),(311,386),(312,387),(313,388),(314,389),(315,390),(316,391),(317,392),(318,393),(319,394),(320,395)], [(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),(321,322,323,324,325),(326,327,328,329,330),(331,332,333,334,335),(336,337,338,339,340),(341,342,343,344,345),(346,347,348,349,350),(351,352,353,354,355),(356,357,358,359,360),(361,362,363,364,365),(366,367,368,369,370),(371,372,373,374,375),(376,377,378,379,380),(381,382,383,384,385),(386,387,388,389,390),(391,392,393,394,395),(396,397,398,399,400)], [(1,44,33,28,38),(2,45,34,29,39),(3,41,35,30,40),(4,42,31,26,36),(5,43,32,27,37),(6,16,400,25,11),(7,17,396,21,12),(8,18,397,22,13),(9,19,398,23,14),(10,20,399,24,15),(46,67,56,51,61),(47,68,57,52,62),(48,69,58,53,63),(49,70,59,54,64),(50,66,60,55,65),(71,92,81,76,86),(72,93,82,77,87),(73,94,83,78,88),(74,95,84,79,89),(75,91,85,80,90),(96,117,106,101,111),(97,118,107,102,112),(98,119,108,103,113),(99,120,109,104,114),(100,116,110,105,115),(121,142,131,126,136),(122,143,132,127,137),(123,144,133,128,138),(124,145,134,129,139),(125,141,135,130,140),(146,167,156,151,161),(147,168,157,152,162),(148,169,158,153,163),(149,170,159,154,164),(150,166,160,155,165),(171,192,181,176,186),(172,193,182,177,187),(173,194,183,178,188),(174,195,184,179,189),(175,191,185,180,190),(196,202,211,216,206),(197,203,212,217,207),(198,204,213,218,208),(199,205,214,219,209),(200,201,215,220,210),(221,230,236,241,231),(222,226,237,242,232),(223,227,238,243,233),(224,228,239,244,234),(225,229,240,245,235),(246,255,261,266,256),(247,251,262,267,257),(248,252,263,268,258),(249,253,264,269,259),(250,254,265,270,260),(271,280,286,291,281),(272,276,287,292,282),(273,277,288,293,283),(274,278,289,294,284),(275,279,290,295,285),(296,305,311,316,306),(297,301,312,317,307),(298,302,313,318,308),(299,303,314,319,309),(300,304,315,320,310),(321,330,336,341,331),(322,326,337,342,332),(323,327,338,343,333),(324,328,339,344,334),(325,329,340,345,335),(346,355,361,366,356),(347,351,362,367,357),(348,352,363,368,358),(349,353,364,369,359),(350,354,365,370,360),(371,380,386,391,381),(372,376,387,392,382),(373,377,388,393,383),(374,378,389,394,384),(375,379,390,395,385)], [(1,98,48,73),(2,99,49,74),(3,100,50,75),(4,96,46,71),(5,97,47,72),(6,345,395,370),(7,341,391,366),(8,342,392,367),(9,343,393,368),(10,344,394,369),(11,340,390,365),(12,336,386,361),(13,337,387,362),(14,338,388,363),(15,339,389,364),(16,335,385,360),(17,331,381,356),(18,332,382,357),(19,333,383,358),(20,334,384,359),(21,330,380,355),(22,326,376,351),(23,327,377,352),(24,328,378,353),(25,329,379,354),(26,101,51,76),(27,102,52,77),(28,103,53,78),(29,104,54,79),(30,105,55,80),(31,106,56,81),(32,107,57,82),(33,108,58,83),(34,109,59,84),(35,110,60,85),(36,111,61,86),(37,112,62,87),(38,113,63,88),(39,114,64,89),(40,115,65,90),(41,116,66,91),(42,117,67,92),(43,118,68,93),(44,119,69,94),(45,120,70,95),(121,196,146,171),(122,197,147,172),(123,198,148,173),(124,199,149,174),(125,200,150,175),(126,216,151,176),(127,217,152,177),(128,218,153,178),(129,219,154,179),(130,220,155,180),(131,211,156,181),(132,212,157,182),(133,213,158,183),(134,214,159,184),(135,215,160,185),(136,206,161,186),(137,207,162,187),(138,208,163,188),(139,209,164,189),(140,210,165,190),(141,201,166,191),(142,202,167,192),(143,203,168,193),(144,204,169,194),(145,205,170,195),(221,271,246,296),(222,272,247,297),(223,273,248,298),(224,274,249,299),(225,275,250,300),(226,276,251,301),(227,277,252,302),(228,278,253,303),(229,279,254,304),(230,280,255,305),(231,281,256,306),(232,282,257,307),(233,283,258,308),(234,284,259,309),(235,285,260,310),(236,286,261,311),(237,287,262,312),(238,288,263,313),(239,289,264,314),(240,290,265,315),(241,291,266,316),(242,292,267,317),(243,293,268,318),(244,294,269,319),(245,295,270,320),(321,371,346,396),(322,372,347,397),(323,373,348,398),(324,374,349,399),(325,375,350,400)], [(1,262,48,237),(2,261,49,236),(3,265,50,240),(4,264,46,239),(5,263,47,238),(6,210,395,190),(7,209,391,189),(8,208,392,188),(9,207,393,187),(10,206,394,186),(11,200,390,175),(12,199,386,174),(13,198,387,173),(14,197,388,172),(15,196,389,171),(16,220,385,180),(17,219,381,179),(18,218,382,178),(19,217,383,177),(20,216,384,176),(21,205,380,195),(22,204,376,194),(23,203,377,193),(24,202,378,192),(25,201,379,191),(26,259,51,234),(27,258,52,233),(28,257,53,232),(29,256,54,231),(30,260,55,235),(31,249,56,224),(32,248,57,223),(33,247,58,222),(34,246,59,221),(35,250,60,225),(36,269,61,244),(37,268,62,243),(38,267,63,242),(39,266,64,241),(40,270,65,245),(41,254,66,229),(42,253,67,228),(43,252,68,227),(44,251,69,226),(45,255,70,230),(71,314,96,289),(72,313,97,288),(73,312,98,287),(74,311,99,286),(75,315,100,290),(76,309,101,284),(77,308,102,283),(78,307,103,282),(79,306,104,281),(80,310,105,285),(81,299,106,274),(82,298,107,273),(83,297,108,272),(84,296,109,271),(85,300,110,275),(86,319,111,294),(87,318,112,293),(88,317,113,292),(89,316,114,291),(90,320,115,295),(91,304,116,279),(92,303,117,278),(93,302,118,277),(94,301,119,276),(95,305,120,280),(121,364,146,339),(122,363,147,338),(123,362,148,337),(124,361,149,336),(125,365,150,340),(126,359,151,334),(127,358,152,333),(128,357,153,332),(129,356,154,331),(130,360,155,335),(131,349,156,324),(132,348,157,323),(133,347,158,322),(134,346,159,321),(135,350,160,325),(136,369,161,344),(137,368,162,343),(138,367,163,342),(139,366,164,341),(140,370,165,345),(141,354,166,329),(142,353,167,328),(143,352,168,327),(144,351,169,326),(145,355,170,330),(181,399,211,374),(182,398,212,373),(183,397,213,372),(184,396,214,371),(185,400,215,375)]])
 

106 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F5A···5L10A···10AJ20A···20AV
order12224444445···510···1020···20
size111122505050502···22···22···2

106 irreducible representations

dim111122222
type++++-+++-
imageC1C2C2C2Q8D5D10D10Dic10
kernelC2×C52⋊4Q8C52⋊4Q8C2×C52⋊6C4C10×C20C5×C10C2×C20C20C2×C10C10
# reps1421212241248

Matrix representation of C2×C52⋊4Q8 ►in GL5(𝔽41)

400000
01000
00100
00010
00001
,
10000
00100
040600
0003535
000640
,
10000
00100
040600
00001
000406
,
10000
01000
00100
0003928
000132
,
10000
040000
035100
000314
0002610

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,40,0,0,0,1,6,0,0,0,0,0,35,6,0,0,0,35,40],[1,0,0,0,0,0,0,40,0,0,0,1,6,0,0,0,0,0,0,40,0,0,0,1,6],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,39,13,0,0,0,28,2],[1,0,0,0,0,0,40,35,0,0,0,0,1,0,0,0,0,0,31,26,0,0,0,4,10] >;
 

C2×C52⋊4Q8 in GAP, Magma, Sage, TeX 

C_2\times C_5^2\rtimes_4Q_8
 
% in TeX
 
G:=Group("C2xC5^2:4Q8");
 
// GroupNames label
 
G:=SmallGroup(400,191);
 
// by ID
 
G=gap.SmallGroup(400,191);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,218,50,1924,11525]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^5=c^5=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
 
// generators/relations
 

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