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G = C5×C65order 325 = 52·13

Abelian group of type [5,65]

direct product, abelian, monomial, 5-elementary

Aliases: C5×C65, SmallGroup(325,2)

Series: Derived Chief Lower central Upper central

C1 — C5×C65
C1C13C65 — C5×C65
C1 — C5×C65
C1 — C5×C65

Generators and relations for C5×C65
 G = < a,b | a5=b65=1, ab=ba >


Smallest permutation representation of C5×C65
Regular action on 325 points
Generators in S325
(1 301 220 135 98)(2 302 221 136 99)(3 303 222 137 100)(4 304 223 138 101)(5 305 224 139 102)(6 306 225 140 103)(7 307 226 141 104)(8 308 227 142 105)(9 309 228 143 106)(10 310 229 144 107)(11 311 230 145 108)(12 312 231 146 109)(13 313 232 147 110)(14 314 233 148 111)(15 315 234 149 112)(16 316 235 150 113)(17 317 236 151 114)(18 318 237 152 115)(19 319 238 153 116)(20 320 239 154 117)(21 321 240 155 118)(22 322 241 156 119)(23 323 242 157 120)(24 324 243 158 121)(25 325 244 159 122)(26 261 245 160 123)(27 262 246 161 124)(28 263 247 162 125)(29 264 248 163 126)(30 265 249 164 127)(31 266 250 165 128)(32 267 251 166 129)(33 268 252 167 130)(34 269 253 168 66)(35 270 254 169 67)(36 271 255 170 68)(37 272 256 171 69)(38 273 257 172 70)(39 274 258 173 71)(40 275 259 174 72)(41 276 260 175 73)(42 277 196 176 74)(43 278 197 177 75)(44 279 198 178 76)(45 280 199 179 77)(46 281 200 180 78)(47 282 201 181 79)(48 283 202 182 80)(49 284 203 183 81)(50 285 204 184 82)(51 286 205 185 83)(52 287 206 186 84)(53 288 207 187 85)(54 289 208 188 86)(55 290 209 189 87)(56 291 210 190 88)(57 292 211 191 89)(58 293 212 192 90)(59 294 213 193 91)(60 295 214 194 92)(61 296 215 195 93)(62 297 216 131 94)(63 298 217 132 95)(64 299 218 133 96)(65 300 219 134 97)
(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)

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

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

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

325 conjugacy classes

class 1 5A···5X13A···13L65A···65KB
order15···513···1365···65
size11···11···11···1

325 irreducible representations

dim1111
type+
imageC1C5C13C65
kernelC5×C65C65C52C5
# reps12412288

Matrix representation of C5×C65 in GL2(𝔽131) generated by

890
01
,
130
043
G:=sub<GL(2,GF(131))| [89,0,0,1],[13,0,0,43] >;

C5×C65 in GAP, Magma, Sage, TeX

C_5\times C_{65}
% in TeX

G:=Group("C5xC65");
// GroupNames label

G:=SmallGroup(325,2);
// by ID

G=gap.SmallGroup(325,2);
# by ID

G:=PCGroup([3,-5,-5,-13]);
// Polycyclic

G:=Group<a,b|a^5=b^65=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C5×C65 in TeX

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