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G = C1048C4order 416 = 25·13

4th semidirect product of C104 and C4 acting via C4/C2=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C1048C4, C83Dic13, C26.9C42, C26.7M4(2), C132C84C4, (C2×C8).8D13, C134(C8⋊C4), C52.62(C2×C4), C4.21(C4×D13), (C2×C4).92D26, (C2×C104).12C2, C2.4(C4×Dic13), C2.2(C8⋊D13), (C4×Dic13).6C2, (C2×Dic13).4C4, C4.13(C2×Dic13), C22.10(C4×D13), (C2×C52).106C22, (C2×C26).31(C2×C4), (C2×C132C8).10C2, SmallGroup(416,22)

Series: Derived Chief Lower central Upper central

C1C26 — C1048C4
C1C13C26C2×C26C2×C52C4×Dic13 — C1048C4
C13C26 — C1048C4
C1C2×C4C2×C8

Generators and relations for C1048C4
 G = < a,b | a104=b4=1, bab-1=a77 >

26C4
26C4
13C8
13C2×C4
13C2×C4
13C8
2Dic13
2Dic13
13C42
13C2×C8
13C8⋊C4

Smallest permutation representation of C1048C4
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)
(1 151 255 328)(2 124 256 405)(3 201 257 378)(4 174 258 351)(5 147 259 324)(6 120 260 401)(7 197 261 374)(8 170 262 347)(9 143 263 320)(10 116 264 397)(11 193 265 370)(12 166 266 343)(13 139 267 316)(14 112 268 393)(15 189 269 366)(16 162 270 339)(17 135 271 416)(18 108 272 389)(19 185 273 362)(20 158 274 335)(21 131 275 412)(22 208 276 385)(23 181 277 358)(24 154 278 331)(25 127 279 408)(26 204 280 381)(27 177 281 354)(28 150 282 327)(29 123 283 404)(30 200 284 377)(31 173 285 350)(32 146 286 323)(33 119 287 400)(34 196 288 373)(35 169 289 346)(36 142 290 319)(37 115 291 396)(38 192 292 369)(39 165 293 342)(40 138 294 315)(41 111 295 392)(42 188 296 365)(43 161 297 338)(44 134 298 415)(45 107 299 388)(46 184 300 361)(47 157 301 334)(48 130 302 411)(49 207 303 384)(50 180 304 357)(51 153 305 330)(52 126 306 407)(53 203 307 380)(54 176 308 353)(55 149 309 326)(56 122 310 403)(57 199 311 376)(58 172 312 349)(59 145 209 322)(60 118 210 399)(61 195 211 372)(62 168 212 345)(63 141 213 318)(64 114 214 395)(65 191 215 368)(66 164 216 341)(67 137 217 314)(68 110 218 391)(69 187 219 364)(70 160 220 337)(71 133 221 414)(72 106 222 387)(73 183 223 360)(74 156 224 333)(75 129 225 410)(76 206 226 383)(77 179 227 356)(78 152 228 329)(79 125 229 406)(80 202 230 379)(81 175 231 352)(82 148 232 325)(83 121 233 402)(84 198 234 375)(85 171 235 348)(86 144 236 321)(87 117 237 398)(88 194 238 371)(89 167 239 344)(90 140 240 317)(91 113 241 394)(92 190 242 367)(93 163 243 340)(94 136 244 313)(95 109 245 390)(96 186 246 363)(97 159 247 336)(98 132 248 413)(99 105 249 386)(100 182 250 359)(101 155 251 332)(102 128 252 409)(103 205 253 382)(104 178 254 355)

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

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

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

116 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H13A···13F26A···26R52A···52X104A···104AV
order1222444444448888888813···1326···2652···52104···104
size11111111262626262222262626262···22···22···22···2

116 irreducible representations

dim11111112222222
type+++++-+
imageC1C2C2C2C4C4C4M4(2)D13Dic13D26C4×D13C4×D13C8⋊D13
kernelC1048C4C2×C132C8C4×Dic13C2×C104C132C8C104C2×Dic13C26C2×C8C8C2×C4C4C22C2
# reps111144446126121248

Matrix representation of C1048C4 in GL4(𝔽313) generated by

2944900
2647700
0024186
0016862
,
124700
1931200
003235
00273310
G:=sub<GL(4,GF(313))| [294,264,0,0,49,77,0,0,0,0,24,168,0,0,186,62],[1,19,0,0,247,312,0,0,0,0,3,273,0,0,235,310] >;

C1048C4 in GAP, Magma, Sage, TeX

C_{104}\rtimes_8C_4
% in TeX

G:=Group("C104:8C4");
// GroupNames label

G:=SmallGroup(416,22);
// by ID

G=gap.SmallGroup(416,22);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,217,55,69,13829]);
// Polycyclic

G:=Group<a,b|a^104=b^4=1,b*a*b^-1=a^77>;
// generators/relations

Export

Subgroup lattice of C1048C4 in TeX

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