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G = C10×C40order 400 = 24·52

Abelian group of type [10,40]

direct product, abelian, monomial

Aliases: C10×C40, SmallGroup(400,111)

Series: Derived Chief Lower central Upper central

C1 — C10×C40
C1C2C4C20C5×C20C5×C40 — C10×C40
C1 — C10×C40
C1 — C10×C40

Generators and relations for C10×C40
 G = < a,b | a10=b40=1, ab=ba >

Subgroups: 88, all normal (14 characteristic)
C1, C2, C2 [×2], C4 [×2], C22, C5 [×6], C8 [×2], C2×C4, C10 [×18], C2×C8, C20 [×12], C2×C10 [×6], C52, C40 [×12], C2×C20 [×6], C5×C10, C5×C10 [×2], C2×C40 [×6], C5×C20 [×2], C102, C5×C40 [×2], C10×C20, C10×C40
Quotients: C1, C2 [×3], C4 [×2], C22, C5 [×6], C8 [×2], C2×C4, C10 [×18], C2×C8, C20 [×12], C2×C10 [×6], C52, C40 [×12], C2×C20 [×6], C5×C10 [×3], C2×C40 [×6], C5×C20 [×2], C102, C5×C40 [×2], C10×C20, C10×C40

Smallest permutation representation of C10×C40
Regular action on 400 points
Generators in S400
(1 386 211 157 318 266 341 189 98 62)(2 387 212 158 319 267 342 190 99 63)(3 388 213 159 320 268 343 191 100 64)(4 389 214 160 281 269 344 192 101 65)(5 390 215 121 282 270 345 193 102 66)(6 391 216 122 283 271 346 194 103 67)(7 392 217 123 284 272 347 195 104 68)(8 393 218 124 285 273 348 196 105 69)(9 394 219 125 286 274 349 197 106 70)(10 395 220 126 287 275 350 198 107 71)(11 396 221 127 288 276 351 199 108 72)(12 397 222 128 289 277 352 200 109 73)(13 398 223 129 290 278 353 161 110 74)(14 399 224 130 291 279 354 162 111 75)(15 400 225 131 292 280 355 163 112 76)(16 361 226 132 293 241 356 164 113 77)(17 362 227 133 294 242 357 165 114 78)(18 363 228 134 295 243 358 166 115 79)(19 364 229 135 296 244 359 167 116 80)(20 365 230 136 297 245 360 168 117 41)(21 366 231 137 298 246 321 169 118 42)(22 367 232 138 299 247 322 170 119 43)(23 368 233 139 300 248 323 171 120 44)(24 369 234 140 301 249 324 172 81 45)(25 370 235 141 302 250 325 173 82 46)(26 371 236 142 303 251 326 174 83 47)(27 372 237 143 304 252 327 175 84 48)(28 373 238 144 305 253 328 176 85 49)(29 374 239 145 306 254 329 177 86 50)(30 375 240 146 307 255 330 178 87 51)(31 376 201 147 308 256 331 179 88 52)(32 377 202 148 309 257 332 180 89 53)(33 378 203 149 310 258 333 181 90 54)(34 379 204 150 311 259 334 182 91 55)(35 380 205 151 312 260 335 183 92 56)(36 381 206 152 313 261 336 184 93 57)(37 382 207 153 314 262 337 185 94 58)(38 383 208 154 315 263 338 186 95 59)(39 384 209 155 316 264 339 187 96 60)(40 385 210 156 317 265 340 188 97 61)
(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)| (1,386,211,157,318,266,341,189,98,62)(2,387,212,158,319,267,342,190,99,63)(3,388,213,159,320,268,343,191,100,64)(4,389,214,160,281,269,344,192,101,65)(5,390,215,121,282,270,345,193,102,66)(6,391,216,122,283,271,346,194,103,67)(7,392,217,123,284,272,347,195,104,68)(8,393,218,124,285,273,348,196,105,69)(9,394,219,125,286,274,349,197,106,70)(10,395,220,126,287,275,350,198,107,71)(11,396,221,127,288,276,351,199,108,72)(12,397,222,128,289,277,352,200,109,73)(13,398,223,129,290,278,353,161,110,74)(14,399,224,130,291,279,354,162,111,75)(15,400,225,131,292,280,355,163,112,76)(16,361,226,132,293,241,356,164,113,77)(17,362,227,133,294,242,357,165,114,78)(18,363,228,134,295,243,358,166,115,79)(19,364,229,135,296,244,359,167,116,80)(20,365,230,136,297,245,360,168,117,41)(21,366,231,137,298,246,321,169,118,42)(22,367,232,138,299,247,322,170,119,43)(23,368,233,139,300,248,323,171,120,44)(24,369,234,140,301,249,324,172,81,45)(25,370,235,141,302,250,325,173,82,46)(26,371,236,142,303,251,326,174,83,47)(27,372,237,143,304,252,327,175,84,48)(28,373,238,144,305,253,328,176,85,49)(29,374,239,145,306,254,329,177,86,50)(30,375,240,146,307,255,330,178,87,51)(31,376,201,147,308,256,331,179,88,52)(32,377,202,148,309,257,332,180,89,53)(33,378,203,149,310,258,333,181,90,54)(34,379,204,150,311,259,334,182,91,55)(35,380,205,151,312,260,335,183,92,56)(36,381,206,152,313,261,336,184,93,57)(37,382,207,153,314,262,337,185,94,58)(38,383,208,154,315,263,338,186,95,59)(39,384,209,155,316,264,339,187,96,60)(40,385,210,156,317,265,340,188,97,61), (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( (1,386,211,157,318,266,341,189,98,62)(2,387,212,158,319,267,342,190,99,63)(3,388,213,159,320,268,343,191,100,64)(4,389,214,160,281,269,344,192,101,65)(5,390,215,121,282,270,345,193,102,66)(6,391,216,122,283,271,346,194,103,67)(7,392,217,123,284,272,347,195,104,68)(8,393,218,124,285,273,348,196,105,69)(9,394,219,125,286,274,349,197,106,70)(10,395,220,126,287,275,350,198,107,71)(11,396,221,127,288,276,351,199,108,72)(12,397,222,128,289,277,352,200,109,73)(13,398,223,129,290,278,353,161,110,74)(14,399,224,130,291,279,354,162,111,75)(15,400,225,131,292,280,355,163,112,76)(16,361,226,132,293,241,356,164,113,77)(17,362,227,133,294,242,357,165,114,78)(18,363,228,134,295,243,358,166,115,79)(19,364,229,135,296,244,359,167,116,80)(20,365,230,136,297,245,360,168,117,41)(21,366,231,137,298,246,321,169,118,42)(22,367,232,138,299,247,322,170,119,43)(23,368,233,139,300,248,323,171,120,44)(24,369,234,140,301,249,324,172,81,45)(25,370,235,141,302,250,325,173,82,46)(26,371,236,142,303,251,326,174,83,47)(27,372,237,143,304,252,327,175,84,48)(28,373,238,144,305,253,328,176,85,49)(29,374,239,145,306,254,329,177,86,50)(30,375,240,146,307,255,330,178,87,51)(31,376,201,147,308,256,331,179,88,52)(32,377,202,148,309,257,332,180,89,53)(33,378,203,149,310,258,333,181,90,54)(34,379,204,150,311,259,334,182,91,55)(35,380,205,151,312,260,335,183,92,56)(36,381,206,152,313,261,336,184,93,57)(37,382,207,153,314,262,337,185,94,58)(38,383,208,154,315,263,338,186,95,59)(39,384,209,155,316,264,339,187,96,60)(40,385,210,156,317,265,340,188,97,61), (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([(1,386,211,157,318,266,341,189,98,62),(2,387,212,158,319,267,342,190,99,63),(3,388,213,159,320,268,343,191,100,64),(4,389,214,160,281,269,344,192,101,65),(5,390,215,121,282,270,345,193,102,66),(6,391,216,122,283,271,346,194,103,67),(7,392,217,123,284,272,347,195,104,68),(8,393,218,124,285,273,348,196,105,69),(9,394,219,125,286,274,349,197,106,70),(10,395,220,126,287,275,350,198,107,71),(11,396,221,127,288,276,351,199,108,72),(12,397,222,128,289,277,352,200,109,73),(13,398,223,129,290,278,353,161,110,74),(14,399,224,130,291,279,354,162,111,75),(15,400,225,131,292,280,355,163,112,76),(16,361,226,132,293,241,356,164,113,77),(17,362,227,133,294,242,357,165,114,78),(18,363,228,134,295,243,358,166,115,79),(19,364,229,135,296,244,359,167,116,80),(20,365,230,136,297,245,360,168,117,41),(21,366,231,137,298,246,321,169,118,42),(22,367,232,138,299,247,322,170,119,43),(23,368,233,139,300,248,323,171,120,44),(24,369,234,140,301,249,324,172,81,45),(25,370,235,141,302,250,325,173,82,46),(26,371,236,142,303,251,326,174,83,47),(27,372,237,143,304,252,327,175,84,48),(28,373,238,144,305,253,328,176,85,49),(29,374,239,145,306,254,329,177,86,50),(30,375,240,146,307,255,330,178,87,51),(31,376,201,147,308,256,331,179,88,52),(32,377,202,148,309,257,332,180,89,53),(33,378,203,149,310,258,333,181,90,54),(34,379,204,150,311,259,334,182,91,55),(35,380,205,151,312,260,335,183,92,56),(36,381,206,152,313,261,336,184,93,57),(37,382,207,153,314,262,337,185,94,58),(38,383,208,154,315,263,338,186,95,59),(39,384,209,155,316,264,339,187,96,60),(40,385,210,156,317,265,340,188,97,61)], [(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 2A2B2C4A4B4C4D5A···5X8A···8H10A···10BT20A···20CR40A···40GJ
order122244445···58···810···1020···2040···40
size111111111···11···11···11···11···1

400 irreducible representations

dim111111111111
type+++
imageC1C2C2C4C4C5C8C10C10C20C20C40
kernelC10×C40C5×C40C10×C20C5×C20C102C2×C40C5×C10C40C2×C20C20C2×C10C10
# reps1212224848244848192

Matrix representation of C10×C40 in GL2(𝔽41) generated by

310
010
,
50
034
G:=sub<GL(2,GF(41))| [31,0,0,10],[5,0,0,34] >;

C10×C40 in GAP, Magma, Sage, TeX

C_{10}\times C_{40}
% in TeX

G:=Group("C10xC40");
// GroupNames label

G:=SmallGroup(400,111);
// by ID

G=gap.SmallGroup(400,111);
# by ID

G:=PCGroup([6,-2,-2,-5,-5,-2,-2,600,88]);
// Polycyclic

G:=Group<a,b|a^10=b^40=1,a*b=b*a>;
// generators/relations

׿
×
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