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## G = C6×C60order 360 = 23·32·5

### Abelian group of type [6,60]

Aliases: C6×C60, SmallGroup(360,115)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6×C60
 Chief series C1 — C2 — C10 — C30 — C3×C30 — C3×C60 — C6×C60
 Lower central C1 — C6×C60
 Upper central 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, C3, C4, C22, C5, C6, C2×C4, C32, C10, C10, C12, C2×C6, C15, C3×C6, C3×C6, C20, C2×C10, C2×C12, C30, C3×C12, C62, C2×C20, C3×C15, C60, C2×C30, C6×C12, C3×C30, C3×C30, C2×C60, C3×C60, C6×C30, C6×C60
Quotients: C1, C2, C3, C4, C22, C5, C6, C2×C4, C32, C10, C12, C2×C6, C15, C3×C6, C20, C2×C10, C2×C12, C30, C3×C12, C62, C2×C20, C3×C15, C60, C2×C30, C6×C12, C3×C30, C2×C60, C3×C60, C6×C30, C6×C60

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

360 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C3 C4 C5 C6 C6 C10 C10 C12 C15 C20 C30 C30 C60 kernel C6×C60 C3×C60 C6×C30 C2×C60 C3×C30 C6×C12 C60 C2×C30 C3×C12 C62 C30 C2×C12 C3×C6 C12 C2×C6 C6 # reps 1 2 1 8 4 4 16 8 8 4 32 32 16 64 32 128

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

 13 0 0 14
,
 31 0 0 30
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

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