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## G = C102.22C22order 400 = 24·52

### 8th non-split extension by C102 of C22 acting via C22/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C102.22C22
 Chief series C1 — C5 — C52 — C5×C10 — C102 — C2×C52⋊6C4 — C102.22C22
 Lower central C52 — C5×C10 — C102.22C22
 Upper central C1 — C22 — C2×C4

Generators and relations for C102.22C22
G = < a,b,c,d | a10=b10=1, c2=b5, d2=a5, ab=ba, cac-1=a-1, ad=da, cbc-1=b-1, bd=db, dcd-1=b5c >

Subgroups: 456 in 104 conjugacy classes, 53 normal (15 characteristic)
C1, C2, C4, C22, C5, C2×C4, C2×C4, C10, C4⋊C4, Dic5, C20, C2×C10, C52, C2×Dic5, C2×C20, C5×C10, C10.D4, C526C4, C526C4, C5×C20, C102, C2×C526C4, C10×C20, C102.22C22
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D5, C4⋊C4, D10, Dic10, C4×D5, C5⋊D4, C5⋊D5, C10.D4, C2×C5⋊D5, C524Q8, C4×C5⋊D5, C527D4, C102.22C22

Smallest permutation representation of C102.22C22
Regular action on 400 points
Generators in S400
```(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 38 151 110 81 169 45 173 117 97)(2 39 152 101 82 170 46 174 118 98)(3 40 153 102 83 161 47 175 119 99)(4 31 154 103 84 162 48 176 120 100)(5 32 155 104 85 163 49 177 111 91)(6 33 156 105 86 164 50 178 112 92)(7 34 157 106 87 165 41 179 113 93)(8 35 158 107 88 166 42 180 114 94)(9 36 159 108 89 167 43 171 115 95)(10 37 160 109 90 168 44 172 116 96)(11 344 80 399 383 64 371 53 337 28)(12 345 71 400 384 65 372 54 338 29)(13 346 72 391 385 66 373 55 339 30)(14 347 73 392 386 67 374 56 340 21)(15 348 74 393 387 68 375 57 331 22)(16 349 75 394 388 69 376 58 332 23)(17 350 76 395 389 70 377 59 333 24)(18 341 77 396 390 61 378 60 334 25)(19 342 78 397 381 62 379 51 335 26)(20 343 79 398 382 63 380 52 336 27)(121 227 149 191 256 134 220 209 189 263)(122 228 150 192 257 135 211 210 190 264)(123 229 141 193 258 136 212 201 181 265)(124 230 142 194 259 137 213 202 182 266)(125 221 143 195 260 138 214 203 183 267)(126 222 144 196 251 139 215 204 184 268)(127 223 145 197 252 140 216 205 185 269)(128 224 146 198 253 131 217 206 186 270)(129 225 147 199 254 132 218 207 187 261)(130 226 148 200 255 133 219 208 188 262)(231 301 314 248 358 293 287 327 275 361)(232 302 315 249 359 294 288 328 276 362)(233 303 316 250 360 295 289 329 277 363)(234 304 317 241 351 296 290 330 278 364)(235 305 318 242 352 297 281 321 279 365)(236 306 319 243 353 298 282 322 280 366)(237 307 320 244 354 299 283 323 271 367)(238 308 311 245 355 300 284 324 272 368)(239 309 312 246 356 291 285 325 273 369)(240 310 313 247 357 292 286 326 274 370)
(1 361 169 358)(2 370 170 357)(3 369 161 356)(4 368 162 355)(5 367 163 354)(6 366 164 353)(7 365 165 352)(8 364 166 351)(9 363 167 360)(10 362 168 359)(11 220 64 227)(12 219 65 226)(13 218 66 225)(14 217 67 224)(15 216 68 223)(16 215 69 222)(17 214 70 221)(18 213 61 230)(19 212 62 229)(20 211 63 228)(21 206 386 146)(22 205 387 145)(23 204 388 144)(24 203 389 143)(25 202 390 142)(26 201 381 141)(27 210 382 150)(28 209 383 149)(29 208 384 148)(30 207 385 147)(31 272 48 245)(32 271 49 244)(33 280 50 243)(34 279 41 242)(35 278 42 241)(36 277 43 250)(37 276 44 249)(38 275 45 248)(39 274 46 247)(40 273 47 246)(51 265 78 258)(52 264 79 257)(53 263 80 256)(54 262 71 255)(55 261 72 254)(56 270 73 253)(57 269 74 252)(58 268 75 251)(59 267 76 260)(60 266 77 259)(81 293 97 231)(82 292 98 240)(83 291 99 239)(84 300 100 238)(85 299 91 237)(86 298 92 236)(87 297 93 235)(88 296 94 234)(89 295 95 233)(90 294 96 232)(101 286 118 310)(102 285 119 309)(103 284 120 308)(104 283 111 307)(105 282 112 306)(106 281 113 305)(107 290 114 304)(108 289 115 303)(109 288 116 302)(110 287 117 301)(121 344 134 371)(122 343 135 380)(123 342 136 379)(124 341 137 378)(125 350 138 377)(126 349 139 376)(127 348 140 375)(128 347 131 374)(129 346 132 373)(130 345 133 372)(151 327 173 314)(152 326 174 313)(153 325 175 312)(154 324 176 311)(155 323 177 320)(156 322 178 319)(157 321 179 318)(158 330 180 317)(159 329 171 316)(160 328 172 315)(181 397 193 335)(182 396 194 334)(183 395 195 333)(184 394 196 332)(185 393 197 331)(186 392 198 340)(187 391 199 339)(188 400 200 338)(189 399 191 337)(190 398 192 336)
(1 257 6 252)(2 258 7 253)(3 259 8 254)(4 260 9 255)(5 251 10 256)(11 323 16 328)(12 324 17 329)(13 325 18 330)(14 326 19 321)(15 327 20 322)(21 286 26 281)(22 287 27 282)(23 288 28 283)(24 289 29 284)(25 290 30 285)(31 138 36 133)(32 139 37 134)(33 140 38 135)(34 131 39 136)(35 132 40 137)(41 128 46 123)(42 129 47 124)(43 130 48 125)(44 121 49 126)(45 122 50 127)(51 352 56 357)(52 353 57 358)(53 354 58 359)(54 355 59 360)(55 356 60 351)(61 317 66 312)(62 318 67 313)(63 319 68 314)(64 320 69 315)(65 311 70 316)(71 368 76 363)(72 369 77 364)(73 370 78 365)(74 361 79 366)(75 362 80 367)(81 190 86 185)(82 181 87 186)(83 182 88 187)(84 183 89 188)(85 184 90 189)(91 196 96 191)(92 197 97 192)(93 198 98 193)(94 199 99 194)(95 200 100 195)(101 201 106 206)(102 202 107 207)(103 203 108 208)(104 204 109 209)(105 205 110 210)(111 144 116 149)(112 145 117 150)(113 146 118 141)(114 147 119 142)(115 148 120 143)(151 211 156 216)(152 212 157 217)(153 213 158 218)(154 214 159 219)(155 215 160 220)(161 266 166 261)(162 267 167 262)(163 268 168 263)(164 269 169 264)(165 270 170 265)(171 226 176 221)(172 227 177 222)(173 228 178 223)(174 229 179 224)(175 230 180 225)(231 398 236 393)(232 399 237 394)(233 400 238 395)(234 391 239 396)(235 392 240 397)(241 373 246 378)(242 374 247 379)(243 375 248 380)(244 376 249 371)(245 377 250 372)(271 349 276 344)(272 350 277 345)(273 341 278 346)(274 342 279 347)(275 343 280 348)(291 334 296 339)(292 335 297 340)(293 336 298 331)(294 337 299 332)(295 338 300 333)(301 382 306 387)(302 383 307 388)(303 384 308 389)(304 385 309 390)(305 386 310 381)```

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

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

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

106 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A ··· 5L 10A ··· 10AJ 20A ··· 20AV order 1 2 2 2 4 4 4 4 4 4 5 ··· 5 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 50 50 50 50 2 ··· 2 2 ··· 2 2 ··· 2

106 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 type + + + + - + + - image C1 C2 C2 C4 D4 Q8 D5 D10 Dic10 C4×D5 C5⋊D4 kernel C102.22C22 C2×C52⋊6C4 C10×C20 C52⋊6C4 C5×C10 C5×C10 C2×C20 C2×C10 C10 C10 C10 # reps 1 2 1 4 1 1 12 12 24 24 24

Matrix representation of C102.22C22 in GL5(𝔽41)

 40 0 0 0 0 0 37 0 0 0 0 0 10 0 0 0 0 0 21 13 0 0 0 2 13
,
 1 0 0 0 0 0 25 0 0 0 0 0 23 0 0 0 0 0 40 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 40 0 0 0 0 0 0 33 33 0 0 0 3 8
,
 9 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 4 13 0 0 0 2 37

`G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,37,0,0,0,0,0,10,0,0,0,0,0,21,2,0,0,0,13,13],[1,0,0,0,0,0,25,0,0,0,0,0,23,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,33,3,0,0,0,33,8],[9,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,4,2,0,0,0,13,37] >;`

C102.22C22 in GAP, Magma, Sage, TeX

`C_{10}^2._{22}C_2^2`
`% in TeX`

`G:=Group("C10^2.22C2^2");`
`// GroupNames label`

`G:=SmallGroup(400,100);`
`// by ID`

`G=gap.SmallGroup(400,100);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,121,31,1924,11525]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^10=b^10=1,c^2=b^5,d^2=a^5,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=b^5*c>;`
`// generators/relations`

׿
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