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G = C1045C4order 416 = 25·13

1st semidirect product of C104 and C4 acting via C4/C2=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C1045C4, C26.4D8, C52.5Q8, C81Dic13, C2.1D104, C26.2Q16, C2.2Dic52, C4.5Dic26, C22.9D52, (C2×C8).3D13, C133(C2.D8), (C2×C104).5C2, C52.55(C2×C4), (C2×C4).69D26, (C2×C26).14D4, C26.13(C4⋊C4), C523C4.3C2, C4.7(C2×Dic13), C2.4(C523C4), (C2×C52).82C22, SmallGroup(416,25)

Series: Derived Chief Lower central Upper central

C1C52 — C1045C4
C1C13C26C2×C26C2×C52C523C4 — C1045C4
C13C26C52 — C1045C4
C1C22C2×C4C2×C8

Generators and relations for C1045C4
 G = < a,b | a104=b4=1, bab-1=a-1 >

52C4
52C4
26C2×C4
26C2×C4
4Dic13
4Dic13
13C4⋊C4
13C4⋊C4
2C2×Dic13
2C2×Dic13
13C2.D8

Smallest permutation representation of C1045C4
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)
(1 121 245 358)(2 120 246 357)(3 119 247 356)(4 118 248 355)(5 117 249 354)(6 116 250 353)(7 115 251 352)(8 114 252 351)(9 113 253 350)(10 112 254 349)(11 111 255 348)(12 110 256 347)(13 109 257 346)(14 108 258 345)(15 107 259 344)(16 106 260 343)(17 105 261 342)(18 208 262 341)(19 207 263 340)(20 206 264 339)(21 205 265 338)(22 204 266 337)(23 203 267 336)(24 202 268 335)(25 201 269 334)(26 200 270 333)(27 199 271 332)(28 198 272 331)(29 197 273 330)(30 196 274 329)(31 195 275 328)(32 194 276 327)(33 193 277 326)(34 192 278 325)(35 191 279 324)(36 190 280 323)(37 189 281 322)(38 188 282 321)(39 187 283 320)(40 186 284 319)(41 185 285 318)(42 184 286 317)(43 183 287 316)(44 182 288 315)(45 181 289 314)(46 180 290 313)(47 179 291 416)(48 178 292 415)(49 177 293 414)(50 176 294 413)(51 175 295 412)(52 174 296 411)(53 173 297 410)(54 172 298 409)(55 171 299 408)(56 170 300 407)(57 169 301 406)(58 168 302 405)(59 167 303 404)(60 166 304 403)(61 165 305 402)(62 164 306 401)(63 163 307 400)(64 162 308 399)(65 161 309 398)(66 160 310 397)(67 159 311 396)(68 158 312 395)(69 157 209 394)(70 156 210 393)(71 155 211 392)(72 154 212 391)(73 153 213 390)(74 152 214 389)(75 151 215 388)(76 150 216 387)(77 149 217 386)(78 148 218 385)(79 147 219 384)(80 146 220 383)(81 145 221 382)(82 144 222 381)(83 143 223 380)(84 142 224 379)(85 141 225 378)(86 140 226 377)(87 139 227 376)(88 138 228 375)(89 137 229 374)(90 136 230 373)(91 135 231 372)(92 134 232 371)(93 133 233 370)(94 132 234 369)(95 131 235 368)(96 130 236 367)(97 129 237 366)(98 128 238 365)(99 127 239 364)(100 126 240 363)(101 125 241 362)(102 124 242 361)(103 123 243 360)(104 122 244 359)

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

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

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

110 conjugacy classes

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

110 irreducible representations

dim111122222222222
type+++-++-+-+-++-
imageC1C2C2C4Q8D4D8Q16D13Dic13D26Dic26D52D104Dic52
kernelC1045C4C523C4C2×C104C104C52C2×C26C26C26C2×C8C8C2×C4C4C22C2C2
# reps12141122612612122424

Matrix representation of C1045C4 in GL3(𝔽313) generated by

100
0334
030928
,
2500
0194175
089119
G:=sub<GL(3,GF(313))| [1,0,0,0,33,309,0,4,28],[25,0,0,0,194,89,0,175,119] >;

C1045C4 in GAP, Magma, Sage, TeX

C_{104}\rtimes_5C_4
% in TeX

G:=Group("C104:5C4");
// GroupNames label

G:=SmallGroup(416,25);
// by ID

G=gap.SmallGroup(416,25);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,151,579,69,13829]);
// Polycyclic

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

Export

Subgroup lattice of C1045C4 in TeX

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