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G = C5×C75order 375 = 3·53

Abelian group of type [5,75]

direct product, abelian, monomial, 5-elementary

Aliases: C5×C75, SmallGroup(375,3)

Series: Derived Chief Lower central Upper central

C1 — C5×C75
C1C5C52C5×C25 — C5×C75
C1 — C5×C75
C1 — C5×C75

Generators and relations for C5×C75
 G = < a,b | a5=b75=1, ab=ba >


Smallest permutation representation of C5×C75
Regular action on 375 points
Generators in S375
(1 166 268 318 95)(2 167 269 319 96)(3 168 270 320 97)(4 169 271 321 98)(5 170 272 322 99)(6 171 273 323 100)(7 172 274 324 101)(8 173 275 325 102)(9 174 276 326 103)(10 175 277 327 104)(11 176 278 328 105)(12 177 279 329 106)(13 178 280 330 107)(14 179 281 331 108)(15 180 282 332 109)(16 181 283 333 110)(17 182 284 334 111)(18 183 285 335 112)(19 184 286 336 113)(20 185 287 337 114)(21 186 288 338 115)(22 187 289 339 116)(23 188 290 340 117)(24 189 291 341 118)(25 190 292 342 119)(26 191 293 343 120)(27 192 294 344 121)(28 193 295 345 122)(29 194 296 346 123)(30 195 297 347 124)(31 196 298 348 125)(32 197 299 349 126)(33 198 300 350 127)(34 199 226 351 128)(35 200 227 352 129)(36 201 228 353 130)(37 202 229 354 131)(38 203 230 355 132)(39 204 231 356 133)(40 205 232 357 134)(41 206 233 358 135)(42 207 234 359 136)(43 208 235 360 137)(44 209 236 361 138)(45 210 237 362 139)(46 211 238 363 140)(47 212 239 364 141)(48 213 240 365 142)(49 214 241 366 143)(50 215 242 367 144)(51 216 243 368 145)(52 217 244 369 146)(53 218 245 370 147)(54 219 246 371 148)(55 220 247 372 149)(56 221 248 373 150)(57 222 249 374 76)(58 223 250 375 77)(59 224 251 301 78)(60 225 252 302 79)(61 151 253 303 80)(62 152 254 304 81)(63 153 255 305 82)(64 154 256 306 83)(65 155 257 307 84)(66 156 258 308 85)(67 157 259 309 86)(68 158 260 310 87)(69 159 261 311 88)(70 160 262 312 89)(71 161 263 313 90)(72 162 264 314 91)(73 163 265 315 92)(74 164 266 316 93)(75 165 267 317 94)
(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)

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

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

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

375 conjugacy classes

class 1 3A3B5A···5X15A···15AV25A···25CV75A···75GR
order1335···515···1525···2575···75
size1111···11···11···11···1

375 irreducible representations

dim11111111
type+
imageC1C3C5C5C15C15C25C75
kernelC5×C75C5×C25C75C5×C15C25C52C15C5
# reps12204408100200

Matrix representation of C5×C75 in GL2(𝔽151) generated by

590
01
,
210
0137
G:=sub<GL(2,GF(151))| [59,0,0,1],[21,0,0,137] >;

C5×C75 in GAP, Magma, Sage, TeX

C_5\times C_{75}
% in TeX

G:=Group("C5xC75");
// GroupNames label

G:=SmallGroup(375,3);
// by ID

G=gap.SmallGroup(375,3);
# by ID

G:=PCGroup([4,-3,-5,-5,-5,205]);
// Polycyclic

G:=Group<a,b|a^5=b^75=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C5×C75 in TeX

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