C5^2:7Q16 - GroupNames

G = C₅²⋊₇Q₁₆ order 400 = 2⁴·5²

2^nd semidirect product of C₅² and Q₁₆ acting via Q₁₆/Q₈=C₂

Aliases: C₅²⋊₇Q₁₆, C₂₀.₁₉D₁₀, Q₈.(C₅⋊D₅), (C₅×Q₈).₅D₅, (C₅×C₁₀).₃₇D₄, C₅⋊₃(C₅⋊Q₁₆), C₅²⋊₇C₈.₂C₂, (Q₈×C₅²).₂C₂, C₅²⋊₄Q₈.₃C₂, C₁₀.₂₅(C₅⋊D₄), (C₅×C₂₀).₁₅C₂², C₂.₇(C₅²⋊₇D₄), C₄.₄(C₂×C₅⋊D₅), SmallGroup(400,106)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅×C₂₀ — C₅²⋊₇Q₁₆

Chief series

C₁ — C₅ — C₅² — C₅×C₁₀ — C₅×C₂₀ — C₅²⋊₄Q₈ — C₅²⋊₇Q₁₆

Lower central

C₅² — C₅×C₁₀ — C₅×C₂₀ — C₅²⋊₇Q₁₆

Upper central

C₁ — C₂ — C₄ — Q₈

Generators and relations for C₅²⋊₇Q₁₆
G = < a,b,c,d | a⁵=b⁵=c⁸=1, d²=c⁴, ab=ba, cac^-1=a^-1, ad=da, cbc^-1=b^-1, bd=db, dcd^-1=c^-1 >

Subgroups: 328 in 72 conjugacy classes, 35 normal (11 characteristic)
C₁, C₂, C₄, C₄, C₅, C₈, Q₈, Q₈, C₁₀, Q₁₆, Dic₅, C₂₀, C₂₀, C₅², C₅⋊₂C₈, Dic₁₀, C₅×Q₈, C₅×C₁₀, C₅⋊Q₁₆, C₅²⋊₆C₄, C₅×C₂₀, C₅×C₂₀, C₅²⋊₇C₈, C₅²⋊₄Q₈, Q₈×C₅², C₅²⋊₇Q₁₆
Quotients: C₁, C₂, C₂², D₄, D₅, Q₁₆, D₁₀, C₅⋊D₄, C₅⋊D₅, C₅⋊Q₁₆, C₂×C₅⋊D₅, C₅²⋊₇D₄, C₅²⋊₇Q₁₆

Smallest permutation representation of C₅²⋊₇Q₁₆
►Regular action on 400 points

Generators in S₄₀₀

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

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

(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)

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

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

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

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

67 conjugacy classes

class	1	2	4A	4B	4C	5A	···	5L	8A	8B	10A	···	10L	20A	···	20AJ
order	1	2	4	4	4	5	···	5	8	8	10	···	10	20	···	20
size	1	1	2	4	100	2	···	2	50	50	2	···	2	4	···	4

67 irreducible representations

dim	1	1	1	1	2	2	2	2	2	4
type	+	+	+	+	+	+	-	+		-
image	C₁	C₂	C₂	C₂	D₄	D₅	Q₁₆	D₁₀	C₅⋊D₄	C₅⋊Q₁₆
kernel	C₅²⋊₇Q₁₆	C₅²⋊₇C₈	C₅²⋊₄Q₈	Q₈×C₅²	C₅×C₁₀	C₅×Q₈	C₅²	C₂₀	C₁₀	C₅
# reps	1	1	1	1	1	12	2	12	24	12

Matrix representation of C₅²⋊₇Q₁₆ ►in GL₆(𝔽₄₁)

7	34	0	0	0	0
7	40	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

34	1	0	0	0	0
40	0	0	0	0	0
0	0	37	0	0	0
0	0	23	10	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	32	0	0	0	0
32	0	0	0	0	0
0	0	31	26	0	0
0	0	23	10	0	0
0	0	0	0	0	26
0	0	0	0	11	17

17	1	0	0	0	0
40	24	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	23	27
0	0	0	0	32	18

G:=sub<GL(6,GF(41))| [7,7,0,0,0,0,34,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[34,40,0,0,0,0,1,0,0,0,0,0,0,0,37,23,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,31,23,0,0,0,0,26,10,0,0,0,0,0,0,0,11,0,0,0,0,26,17],[17,40,0,0,0,0,1,24,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,23,32,0,0,0,0,27,18] >;

C₅²⋊₇Q₁₆ in GAP, Magma, Sage, TeX

C_5^2\rtimes_7Q_{16}

% in TeX

G:=Group("C5^2:7Q16");

// GroupNames label

G:=SmallGroup(400,106);

// by ID

G=gap.SmallGroup(400,106);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,73,55,218,116,50,1924,11525]);

// Polycyclic

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

// generators/relations

G = C52⋊7Q16 order 400 = 24·52

2nd semidirect product of C52 and Q16 acting via Q16/Q8=C2

G = C₅²⋊₇Q₁₆ order 400 = 2⁴·5²

2^nd semidirect product of C₅² and Q₁₆ acting via Q₁₆/Q₈=C₂