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G = C5×C70order 350 = 2·52·7

Abelian group of type [5,70]

direct product, abelian, monomial, 5-elementary

Aliases: C5×C70, SmallGroup(350,10)

Series: Derived Chief Lower central Upper central

C1 — C5×C70
C1C7C35C5×C35 — C5×C70
C1 — C5×C70
C1 — C5×C70

Generators and relations for C5×C70
 G = < a,b | a5=b70=1, ab=ba >


Smallest permutation representation of C5×C70
Regular action on 350 points
Generators in S350
(1 97 156 314 241)(2 98 157 315 242)(3 99 158 316 243)(4 100 159 317 244)(5 101 160 318 245)(6 102 161 319 246)(7 103 162 320 247)(8 104 163 321 248)(9 105 164 322 249)(10 106 165 323 250)(11 107 166 324 251)(12 108 167 325 252)(13 109 168 326 253)(14 110 169 327 254)(15 111 170 328 255)(16 112 171 329 256)(17 113 172 330 257)(18 114 173 331 258)(19 115 174 332 259)(20 116 175 333 260)(21 117 176 334 261)(22 118 177 335 262)(23 119 178 336 263)(24 120 179 337 264)(25 121 180 338 265)(26 122 181 339 266)(27 123 182 340 267)(28 124 183 341 268)(29 125 184 342 269)(30 126 185 343 270)(31 127 186 344 271)(32 128 187 345 272)(33 129 188 346 273)(34 130 189 347 274)(35 131 190 348 275)(36 132 191 349 276)(37 133 192 350 277)(38 134 193 281 278)(39 135 194 282 279)(40 136 195 283 280)(41 137 196 284 211)(42 138 197 285 212)(43 139 198 286 213)(44 140 199 287 214)(45 71 200 288 215)(46 72 201 289 216)(47 73 202 290 217)(48 74 203 291 218)(49 75 204 292 219)(50 76 205 293 220)(51 77 206 294 221)(52 78 207 295 222)(53 79 208 296 223)(54 80 209 297 224)(55 81 210 298 225)(56 82 141 299 226)(57 83 142 300 227)(58 84 143 301 228)(59 85 144 302 229)(60 86 145 303 230)(61 87 146 304 231)(62 88 147 305 232)(63 89 148 306 233)(64 90 149 307 234)(65 91 150 308 235)(66 92 151 309 236)(67 93 152 310 237)(68 94 153 311 238)(69 95 154 312 239)(70 96 155 313 240)
(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)

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

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

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

350 conjugacy classes

class 1  2 5A···5X7A···7F10A···10X14A···14F35A···35EN70A···70EN
order125···57···710···1014···1435···3570···70
size111···11···11···11···11···11···1

350 irreducible representations

dim11111111
type++
imageC1C2C5C7C10C14C35C70
kernelC5×C70C5×C35C70C5×C10C35C52C10C5
# reps11246246144144

Matrix representation of C5×C70 in GL2(𝔽71) generated by

10
025
,
460
051
G:=sub<GL(2,GF(71))| [1,0,0,25],[46,0,0,51] >;

C5×C70 in GAP, Magma, Sage, TeX

C_5\times C_{70}
% in TeX

G:=Group("C5xC70");
// GroupNames label

G:=SmallGroup(350,10);
// by ID

G=gap.SmallGroup(350,10);
# by ID

G:=PCGroup([4,-2,-5,-5,-7]);
// Polycyclic

G:=Group<a,b|a^5=b^70=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C5×C70 in TeX

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