Copied to
clipboard

G = C6×C60order 360 = 23·32·5

Abelian group of type [6,60]

direct product, abelian, monomial

Aliases: C6×C60, SmallGroup(360,115)

Series: Derived Chief Lower central Upper central

C1 — C6×C60
C1C2C10C30C3×C30C3×C60 — C6×C60
C1 — C6×C60
C1 — C6×C60

Generators and relations for C6×C60
 G = < a,b | a6=b60=1, ab=ba >

Subgroups: 96, all normal (16 characteristic)
C1, C2, C2 [×2], C3 [×4], C4 [×2], C22, C5, C6 [×12], C2×C4, C32, C10, C10 [×2], C12 [×8], C2×C6 [×4], C15 [×4], C3×C6, C3×C6 [×2], C20 [×2], C2×C10, C2×C12 [×4], C30 [×12], C3×C12 [×2], C62, C2×C20, C3×C15, C60 [×8], C2×C30 [×4], C6×C12, C3×C30, C3×C30 [×2], C2×C60 [×4], C3×C60 [×2], C6×C30, C6×C60
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, C5, C6 [×12], C2×C4, C32, C10 [×3], C12 [×8], C2×C6 [×4], C15 [×4], C3×C6 [×3], C20 [×2], C2×C10, C2×C12 [×4], C30 [×12], C3×C12 [×2], C62, C2×C20, C3×C15, C60 [×8], C2×C30 [×4], C6×C12, C3×C30 [×3], C2×C60 [×4], C3×C60 [×2], C6×C30, C6×C60

Smallest permutation representation of C6×C60
Regular action on 360 points
Generators in S360
(1 76 357 275 157 209)(2 77 358 276 158 210)(3 78 359 277 159 211)(4 79 360 278 160 212)(5 80 301 279 161 213)(6 81 302 280 162 214)(7 82 303 281 163 215)(8 83 304 282 164 216)(9 84 305 283 165 217)(10 85 306 284 166 218)(11 86 307 285 167 219)(12 87 308 286 168 220)(13 88 309 287 169 221)(14 89 310 288 170 222)(15 90 311 289 171 223)(16 91 312 290 172 224)(17 92 313 291 173 225)(18 93 314 292 174 226)(19 94 315 293 175 227)(20 95 316 294 176 228)(21 96 317 295 177 229)(22 97 318 296 178 230)(23 98 319 297 179 231)(24 99 320 298 180 232)(25 100 321 299 121 233)(26 101 322 300 122 234)(27 102 323 241 123 235)(28 103 324 242 124 236)(29 104 325 243 125 237)(30 105 326 244 126 238)(31 106 327 245 127 239)(32 107 328 246 128 240)(33 108 329 247 129 181)(34 109 330 248 130 182)(35 110 331 249 131 183)(36 111 332 250 132 184)(37 112 333 251 133 185)(38 113 334 252 134 186)(39 114 335 253 135 187)(40 115 336 254 136 188)(41 116 337 255 137 189)(42 117 338 256 138 190)(43 118 339 257 139 191)(44 119 340 258 140 192)(45 120 341 259 141 193)(46 61 342 260 142 194)(47 62 343 261 143 195)(48 63 344 262 144 196)(49 64 345 263 145 197)(50 65 346 264 146 198)(51 66 347 265 147 199)(52 67 348 266 148 200)(53 68 349 267 149 201)(54 69 350 268 150 202)(55 70 351 269 151 203)(56 71 352 270 152 204)(57 72 353 271 153 205)(58 73 354 272 154 206)(59 74 355 273 155 207)(60 75 356 274 156 208)
(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)

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

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

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

360 conjugacy classes

class 1 2A2B2C3A···3H4A4B4C4D5A5B5C5D6A···6X10A···10L12A···12AF15A···15AF20A···20P30A···30CR60A···60DX
order12223···3444455556···610···1012···1215···1520···2030···3060···60
size11111···1111111111···11···11···11···11···11···11···1

360 irreducible representations

dim1111111111111111
type+++
imageC1C2C2C3C4C5C6C6C10C10C12C15C20C30C30C60
kernelC6×C60C3×C60C6×C30C2×C60C3×C30C6×C12C60C2×C30C3×C12C62C30C2×C12C3×C6C12C2×C6C6
# reps121844168843232166432128

Matrix representation of C6×C60 in GL2(𝔽61) generated by

130
014
,
310
030
G:=sub<GL(2,GF(61))| [13,0,0,14],[31,0,0,30] >;

C6×C60 in GAP, Magma, Sage, TeX

C_6\times C_{60}
% in TeX

G:=Group("C6xC60");
// GroupNames label

G:=SmallGroup(360,115);
// by ID

G=gap.SmallGroup(360,115);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-5,-2,1080]);
// Polycyclic

G:=Group<a,b|a^6=b^60=1,a*b=b*a>;
// generators/relations

׿
×
𝔽