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G = C5×C60order 300 = 22·3·52

Abelian group of type [5,60]

direct product, abelian, monomial, 5-elementary

Aliases: C5×C60, SmallGroup(300,21)

Series: Derived Chief Lower central Upper central

C1 — C5×C60
C1C2C10C5×C10C5×C30 — C5×C60
C1 — C5×C60
C1 — C5×C60

Generators and relations for C5×C60
 G = < a,b | a5=b60=1, ab=ba >


Smallest permutation representation of C5×C60
Regular action on 300 points
Generators in S300
(1 96 169 276 240)(2 97 170 277 181)(3 98 171 278 182)(4 99 172 279 183)(5 100 173 280 184)(6 101 174 281 185)(7 102 175 282 186)(8 103 176 283 187)(9 104 177 284 188)(10 105 178 285 189)(11 106 179 286 190)(12 107 180 287 191)(13 108 121 288 192)(14 109 122 289 193)(15 110 123 290 194)(16 111 124 291 195)(17 112 125 292 196)(18 113 126 293 197)(19 114 127 294 198)(20 115 128 295 199)(21 116 129 296 200)(22 117 130 297 201)(23 118 131 298 202)(24 119 132 299 203)(25 120 133 300 204)(26 61 134 241 205)(27 62 135 242 206)(28 63 136 243 207)(29 64 137 244 208)(30 65 138 245 209)(31 66 139 246 210)(32 67 140 247 211)(33 68 141 248 212)(34 69 142 249 213)(35 70 143 250 214)(36 71 144 251 215)(37 72 145 252 216)(38 73 146 253 217)(39 74 147 254 218)(40 75 148 255 219)(41 76 149 256 220)(42 77 150 257 221)(43 78 151 258 222)(44 79 152 259 223)(45 80 153 260 224)(46 81 154 261 225)(47 82 155 262 226)(48 83 156 263 227)(49 84 157 264 228)(50 85 158 265 229)(51 86 159 266 230)(52 87 160 267 231)(53 88 161 268 232)(54 89 162 269 233)(55 90 163 270 234)(56 91 164 271 235)(57 92 165 272 236)(58 93 166 273 237)(59 94 167 274 238)(60 95 168 275 239)
(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)

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

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

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

300 conjugacy classes

class 1  2 3A3B4A4B5A···5X6A6B10A···10X12A12B12C12D15A···15AV20A···20AV30A···30AV60A···60CR
order1233445···56610···101212121215···1520···2030···3060···60
size1111111···1111···111111···11···11···11···1

300 irreducible representations

dim111111111111
type++
imageC1C2C3C4C5C6C10C12C15C20C30C60
kernelC5×C60C5×C30C5×C20C5×C15C60C5×C10C30C52C20C15C10C5
# reps112224224448484896

Matrix representation of C5×C60 in GL2(𝔽61) generated by

340
020
,
390
050
G:=sub<GL(2,GF(61))| [34,0,0,20],[39,0,0,50] >;

C5×C60 in GAP, Magma, Sage, TeX

C_5\times C_{60}
% in TeX

G:=Group("C5xC60");
// GroupNames label

G:=SmallGroup(300,21);
// by ID

G=gap.SmallGroup(300,21);
# by ID

G:=PCGroup([5,-2,-3,-5,-5,-2,750]);
// Polycyclic

G:=Group<a,b|a^5=b^60=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C5×C60 in TeX

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