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G = C1046C4order 416 = 25·13

2nd semidirect product of C104 and C4 acting via C4/C2=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C1046C4, C52.4Q8, C82Dic13, C4.4Dic26, C26.2SD16, C22.8D52, (C2×C8).6D13, C133(C4.Q8), C52.54(C2×C4), (C2×C104).8C2, (C2×C26).13D4, (C2×C4).68D26, C26.12(C4⋊C4), C523C4.2C2, C4.6(C2×Dic13), C2.3(C523C4), C2.2(C104⋊C2), (C2×C52).81C22, SmallGroup(416,24)

Series: Derived Chief Lower central Upper central

C1C52 — C1046C4
C1C13C26C2×C26C2×C52C523C4 — C1046C4
C13C26C52 — C1046C4
C1C22C2×C4C2×C8

Generators and relations for C1046C4
 G = < a,b | a104=b4=1, bab-1=a51 >

52C4
52C4
26C2×C4
26C2×C4
4Dic13
4Dic13
13C4⋊C4
13C4⋊C4
2C2×Dic13
2C2×Dic13
13C4.Q8

Smallest permutation representation of C1046C4
Regular action on 416 points
Generators in S416
(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)(313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416)
(1 406 181 238)(2 353 182 289)(3 404 183 236)(4 351 184 287)(5 402 185 234)(6 349 186 285)(7 400 187 232)(8 347 188 283)(9 398 189 230)(10 345 190 281)(11 396 191 228)(12 343 192 279)(13 394 193 226)(14 341 194 277)(15 392 195 224)(16 339 196 275)(17 390 197 222)(18 337 198 273)(19 388 199 220)(20 335 200 271)(21 386 201 218)(22 333 202 269)(23 384 203 216)(24 331 204 267)(25 382 205 214)(26 329 206 265)(27 380 207 212)(28 327 208 263)(29 378 105 210)(30 325 106 261)(31 376 107 312)(32 323 108 259)(33 374 109 310)(34 321 110 257)(35 372 111 308)(36 319 112 255)(37 370 113 306)(38 317 114 253)(39 368 115 304)(40 315 116 251)(41 366 117 302)(42 313 118 249)(43 364 119 300)(44 415 120 247)(45 362 121 298)(46 413 122 245)(47 360 123 296)(48 411 124 243)(49 358 125 294)(50 409 126 241)(51 356 127 292)(52 407 128 239)(53 354 129 290)(54 405 130 237)(55 352 131 288)(56 403 132 235)(57 350 133 286)(58 401 134 233)(59 348 135 284)(60 399 136 231)(61 346 137 282)(62 397 138 229)(63 344 139 280)(64 395 140 227)(65 342 141 278)(66 393 142 225)(67 340 143 276)(68 391 144 223)(69 338 145 274)(70 389 146 221)(71 336 147 272)(72 387 148 219)(73 334 149 270)(74 385 150 217)(75 332 151 268)(76 383 152 215)(77 330 153 266)(78 381 154 213)(79 328 155 264)(80 379 156 211)(81 326 157 262)(82 377 158 209)(83 324 159 260)(84 375 160 311)(85 322 161 258)(86 373 162 309)(87 320 163 256)(88 371 164 307)(89 318 165 254)(90 369 166 305)(91 316 167 252)(92 367 168 303)(93 314 169 250)(94 365 170 301)(95 416 171 248)(96 363 172 299)(97 414 173 246)(98 361 174 297)(99 412 175 244)(100 359 176 295)(101 410 177 242)(102 357 178 293)(103 408 179 240)(104 355 180 291)

G:=sub<Sym(416)| (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)(313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416), (1,406,181,238)(2,353,182,289)(3,404,183,236)(4,351,184,287)(5,402,185,234)(6,349,186,285)(7,400,187,232)(8,347,188,283)(9,398,189,230)(10,345,190,281)(11,396,191,228)(12,343,192,279)(13,394,193,226)(14,341,194,277)(15,392,195,224)(16,339,196,275)(17,390,197,222)(18,337,198,273)(19,388,199,220)(20,335,200,271)(21,386,201,218)(22,333,202,269)(23,384,203,216)(24,331,204,267)(25,382,205,214)(26,329,206,265)(27,380,207,212)(28,327,208,263)(29,378,105,210)(30,325,106,261)(31,376,107,312)(32,323,108,259)(33,374,109,310)(34,321,110,257)(35,372,111,308)(36,319,112,255)(37,370,113,306)(38,317,114,253)(39,368,115,304)(40,315,116,251)(41,366,117,302)(42,313,118,249)(43,364,119,300)(44,415,120,247)(45,362,121,298)(46,413,122,245)(47,360,123,296)(48,411,124,243)(49,358,125,294)(50,409,126,241)(51,356,127,292)(52,407,128,239)(53,354,129,290)(54,405,130,237)(55,352,131,288)(56,403,132,235)(57,350,133,286)(58,401,134,233)(59,348,135,284)(60,399,136,231)(61,346,137,282)(62,397,138,229)(63,344,139,280)(64,395,140,227)(65,342,141,278)(66,393,142,225)(67,340,143,276)(68,391,144,223)(69,338,145,274)(70,389,146,221)(71,336,147,272)(72,387,148,219)(73,334,149,270)(74,385,150,217)(75,332,151,268)(76,383,152,215)(77,330,153,266)(78,381,154,213)(79,328,155,264)(80,379,156,211)(81,326,157,262)(82,377,158,209)(83,324,159,260)(84,375,160,311)(85,322,161,258)(86,373,162,309)(87,320,163,256)(88,371,164,307)(89,318,165,254)(90,369,166,305)(91,316,167,252)(92,367,168,303)(93,314,169,250)(94,365,170,301)(95,416,171,248)(96,363,172,299)(97,414,173,246)(98,361,174,297)(99,412,175,244)(100,359,176,295)(101,410,177,242)(102,357,178,293)(103,408,179,240)(104,355,180,291)>;

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)(313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416), (1,406,181,238)(2,353,182,289)(3,404,183,236)(4,351,184,287)(5,402,185,234)(6,349,186,285)(7,400,187,232)(8,347,188,283)(9,398,189,230)(10,345,190,281)(11,396,191,228)(12,343,192,279)(13,394,193,226)(14,341,194,277)(15,392,195,224)(16,339,196,275)(17,390,197,222)(18,337,198,273)(19,388,199,220)(20,335,200,271)(21,386,201,218)(22,333,202,269)(23,384,203,216)(24,331,204,267)(25,382,205,214)(26,329,206,265)(27,380,207,212)(28,327,208,263)(29,378,105,210)(30,325,106,261)(31,376,107,312)(32,323,108,259)(33,374,109,310)(34,321,110,257)(35,372,111,308)(36,319,112,255)(37,370,113,306)(38,317,114,253)(39,368,115,304)(40,315,116,251)(41,366,117,302)(42,313,118,249)(43,364,119,300)(44,415,120,247)(45,362,121,298)(46,413,122,245)(47,360,123,296)(48,411,124,243)(49,358,125,294)(50,409,126,241)(51,356,127,292)(52,407,128,239)(53,354,129,290)(54,405,130,237)(55,352,131,288)(56,403,132,235)(57,350,133,286)(58,401,134,233)(59,348,135,284)(60,399,136,231)(61,346,137,282)(62,397,138,229)(63,344,139,280)(64,395,140,227)(65,342,141,278)(66,393,142,225)(67,340,143,276)(68,391,144,223)(69,338,145,274)(70,389,146,221)(71,336,147,272)(72,387,148,219)(73,334,149,270)(74,385,150,217)(75,332,151,268)(76,383,152,215)(77,330,153,266)(78,381,154,213)(79,328,155,264)(80,379,156,211)(81,326,157,262)(82,377,158,209)(83,324,159,260)(84,375,160,311)(85,322,161,258)(86,373,162,309)(87,320,163,256)(88,371,164,307)(89,318,165,254)(90,369,166,305)(91,316,167,252)(92,367,168,303)(93,314,169,250)(94,365,170,301)(95,416,171,248)(96,363,172,299)(97,414,173,246)(98,361,174,297)(99,412,175,244)(100,359,176,295)(101,410,177,242)(102,357,178,293)(103,408,179,240)(104,355,180,291) );

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),(313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416)], [(1,406,181,238),(2,353,182,289),(3,404,183,236),(4,351,184,287),(5,402,185,234),(6,349,186,285),(7,400,187,232),(8,347,188,283),(9,398,189,230),(10,345,190,281),(11,396,191,228),(12,343,192,279),(13,394,193,226),(14,341,194,277),(15,392,195,224),(16,339,196,275),(17,390,197,222),(18,337,198,273),(19,388,199,220),(20,335,200,271),(21,386,201,218),(22,333,202,269),(23,384,203,216),(24,331,204,267),(25,382,205,214),(26,329,206,265),(27,380,207,212),(28,327,208,263),(29,378,105,210),(30,325,106,261),(31,376,107,312),(32,323,108,259),(33,374,109,310),(34,321,110,257),(35,372,111,308),(36,319,112,255),(37,370,113,306),(38,317,114,253),(39,368,115,304),(40,315,116,251),(41,366,117,302),(42,313,118,249),(43,364,119,300),(44,415,120,247),(45,362,121,298),(46,413,122,245),(47,360,123,296),(48,411,124,243),(49,358,125,294),(50,409,126,241),(51,356,127,292),(52,407,128,239),(53,354,129,290),(54,405,130,237),(55,352,131,288),(56,403,132,235),(57,350,133,286),(58,401,134,233),(59,348,135,284),(60,399,136,231),(61,346,137,282),(62,397,138,229),(63,344,139,280),(64,395,140,227),(65,342,141,278),(66,393,142,225),(67,340,143,276),(68,391,144,223),(69,338,145,274),(70,389,146,221),(71,336,147,272),(72,387,148,219),(73,334,149,270),(74,385,150,217),(75,332,151,268),(76,383,152,215),(77,330,153,266),(78,381,154,213),(79,328,155,264),(80,379,156,211),(81,326,157,262),(82,377,158,209),(83,324,159,260),(84,375,160,311),(85,322,161,258),(86,373,162,309),(87,320,163,256),(88,371,164,307),(89,318,165,254),(90,369,166,305),(91,316,167,252),(92,367,168,303),(93,314,169,250),(94,365,170,301),(95,416,171,248),(96,363,172,299),(97,414,173,246),(98,361,174,297),(99,412,175,244),(100,359,176,295),(101,410,177,242),(102,357,178,293),(103,408,179,240),(104,355,180,291)])

110 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F8A8B8C8D13A···13F26A···26R52A···52X104A···104AV
order1222444444888813···1326···2652···52104···104
size1111225252525222222···22···22···22···2

110 irreducible representations

dim1111222222222
type+++-++-+-+
imageC1C2C2C4Q8D4SD16D13Dic13D26Dic26D52C104⋊C2
kernelC1046C4C523C4C2×C104C104C52C2×C26C26C2×C8C8C2×C4C4C22C2
# reps12141146126121248

Matrix representation of C1046C4 in GL3(𝔽313) generated by

31200
013028
02856
,
2500
020041
0139113
G:=sub<GL(3,GF(313))| [312,0,0,0,130,285,0,28,6],[25,0,0,0,200,139,0,41,113] >;

C1046C4 in GAP, Magma, Sage, TeX

C_{104}\rtimes_6C_4
% in TeX

G:=Group("C104:6C4");
// GroupNames label

G:=SmallGroup(416,24);
// by ID

G=gap.SmallGroup(416,24);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,55,579,69,13829]);
// Polycyclic

G:=Group<a,b|a^104=b^4=1,b*a*b^-1=a^51>;
// generators/relations

Export

Subgroup lattice of C1046C4 in TeX

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