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G = C4×C104order 416 = 25·13

Abelian group of type [4,104]

direct product, abelian, monomial, 2-elementary

Aliases: C4×C104, SmallGroup(416,46)

Series: Derived Chief Lower central Upper central

C1 — C4×C104
C1C2C22C2×C4C2×C52C2×C104 — C4×C104
C1 — C4×C104
C1 — C4×C104

Generators and relations for C4×C104
 G = < a,b | a4=b104=1, ab=ba >


Smallest permutation representation of C4×C104
Regular action on 416 points
Generators in S416
(1 308 197 342)(2 309 198 343)(3 310 199 344)(4 311 200 345)(5 312 201 346)(6 209 202 347)(7 210 203 348)(8 211 204 349)(9 212 205 350)(10 213 206 351)(11 214 207 352)(12 215 208 353)(13 216 105 354)(14 217 106 355)(15 218 107 356)(16 219 108 357)(17 220 109 358)(18 221 110 359)(19 222 111 360)(20 223 112 361)(21 224 113 362)(22 225 114 363)(23 226 115 364)(24 227 116 365)(25 228 117 366)(26 229 118 367)(27 230 119 368)(28 231 120 369)(29 232 121 370)(30 233 122 371)(31 234 123 372)(32 235 124 373)(33 236 125 374)(34 237 126 375)(35 238 127 376)(36 239 128 377)(37 240 129 378)(38 241 130 379)(39 242 131 380)(40 243 132 381)(41 244 133 382)(42 245 134 383)(43 246 135 384)(44 247 136 385)(45 248 137 386)(46 249 138 387)(47 250 139 388)(48 251 140 389)(49 252 141 390)(50 253 142 391)(51 254 143 392)(52 255 144 393)(53 256 145 394)(54 257 146 395)(55 258 147 396)(56 259 148 397)(57 260 149 398)(58 261 150 399)(59 262 151 400)(60 263 152 401)(61 264 153 402)(62 265 154 403)(63 266 155 404)(64 267 156 405)(65 268 157 406)(66 269 158 407)(67 270 159 408)(68 271 160 409)(69 272 161 410)(70 273 162 411)(71 274 163 412)(72 275 164 413)(73 276 165 414)(74 277 166 415)(75 278 167 416)(76 279 168 313)(77 280 169 314)(78 281 170 315)(79 282 171 316)(80 283 172 317)(81 284 173 318)(82 285 174 319)(83 286 175 320)(84 287 176 321)(85 288 177 322)(86 289 178 323)(87 290 179 324)(88 291 180 325)(89 292 181 326)(90 293 182 327)(91 294 183 328)(92 295 184 329)(93 296 185 330)(94 297 186 331)(95 298 187 332)(96 299 188 333)(97 300 189 334)(98 301 190 335)(99 302 191 336)(100 303 192 337)(101 304 193 338)(102 305 194 339)(103 306 195 340)(104 307 196 341)
(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)

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

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

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

416 conjugacy classes

class 1 2A2B2C4A···4L8A···8P13A···13L26A···26AJ52A···52EN104A···104GJ
order12224···48···813···1326···2652···52104···104
size11111···11···11···11···11···11···1

416 irreducible representations

dim111111111111
type+++
imageC1C2C2C4C4C8C13C26C26C52C52C104
kernelC4×C104C4×C52C2×C104C104C2×C52C52C4×C8C42C2×C8C8C2×C4C4
# reps11284161212249648192

Matrix representation of C4×C104 in GL2(𝔽313) generated by

250
0312
,
1630
0133
G:=sub<GL(2,GF(313))| [25,0,0,312],[163,0,0,133] >;

C4×C104 in GAP, Magma, Sage, TeX

C_4\times C_{104}
% in TeX

G:=Group("C4xC104");
// GroupNames label

G:=SmallGroup(416,46);
// by ID

G=gap.SmallGroup(416,46);
# by ID

G:=PCGroup([6,-2,-2,-13,-2,-2,-2,312,631,117]);
// Polycyclic

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

Export

Subgroup lattice of C4×C104 in TeX

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