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G = C4×C100order 400 = 24·52

Abelian group of type [4,100]

direct product, abelian, monomial, 2-elementary

Aliases: C4×C100, SmallGroup(400,20)

Series: Derived Chief Lower central Upper central

C1 — C4×C100
C1C5C10C2×C10C2×C50C2×C100 — C4×C100
C1 — C4×C100
C1 — C4×C100

Generators and relations for C4×C100
 G = < a,b | a4=b100=1, ab=ba >


Smallest permutation representation of C4×C100
Regular action on 400 points
Generators in S400
(1 334 188 299)(2 335 189 300)(3 336 190 201)(4 337 191 202)(5 338 192 203)(6 339 193 204)(7 340 194 205)(8 341 195 206)(9 342 196 207)(10 343 197 208)(11 344 198 209)(12 345 199 210)(13 346 200 211)(14 347 101 212)(15 348 102 213)(16 349 103 214)(17 350 104 215)(18 351 105 216)(19 352 106 217)(20 353 107 218)(21 354 108 219)(22 355 109 220)(23 356 110 221)(24 357 111 222)(25 358 112 223)(26 359 113 224)(27 360 114 225)(28 361 115 226)(29 362 116 227)(30 363 117 228)(31 364 118 229)(32 365 119 230)(33 366 120 231)(34 367 121 232)(35 368 122 233)(36 369 123 234)(37 370 124 235)(38 371 125 236)(39 372 126 237)(40 373 127 238)(41 374 128 239)(42 375 129 240)(43 376 130 241)(44 377 131 242)(45 378 132 243)(46 379 133 244)(47 380 134 245)(48 381 135 246)(49 382 136 247)(50 383 137 248)(51 384 138 249)(52 385 139 250)(53 386 140 251)(54 387 141 252)(55 388 142 253)(56 389 143 254)(57 390 144 255)(58 391 145 256)(59 392 146 257)(60 393 147 258)(61 394 148 259)(62 395 149 260)(63 396 150 261)(64 397 151 262)(65 398 152 263)(66 399 153 264)(67 400 154 265)(68 301 155 266)(69 302 156 267)(70 303 157 268)(71 304 158 269)(72 305 159 270)(73 306 160 271)(74 307 161 272)(75 308 162 273)(76 309 163 274)(77 310 164 275)(78 311 165 276)(79 312 166 277)(80 313 167 278)(81 314 168 279)(82 315 169 280)(83 316 170 281)(84 317 171 282)(85 318 172 283)(86 319 173 284)(87 320 174 285)(88 321 175 286)(89 322 176 287)(90 323 177 288)(91 324 178 289)(92 325 179 290)(93 326 180 291)(94 327 181 292)(95 328 182 293)(96 329 183 294)(97 330 184 295)(98 331 185 296)(99 332 186 297)(100 333 187 298)
(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)

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

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

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

400 conjugacy classes

class 1 2A2B2C4A···4L5A5B5C5D10A···10L20A···20AV25A···25T50A···50BH100A···100IF
order12224···4555510···1020···2025···2550···50100···100
size11111···111111···11···11···11···11···1

400 irreducible representations

dim111111111
type++
imageC1C2C4C5C10C20C25C50C100
kernelC4×C100C2×C100C100C4×C20C2×C20C20C42C2×C4C4
# reps1312412482060240

Matrix representation of C4×C100 in GL2(𝔽101) generated by

10
091
,
290
073
G:=sub<GL(2,GF(101))| [1,0,0,91],[29,0,0,73] >;

C4×C100 in GAP, Magma, Sage, TeX

C_4\times C_{100}
% in TeX

G:=Group("C4xC100");
// GroupNames label

G:=SmallGroup(400,20);
// by ID

G=gap.SmallGroup(400,20);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,120,247,374]);
// Polycyclic

G:=Group<a,b|a^4=b^100=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C4×C100 in TeX

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