metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C45⋊3C8, C90.3C4, C60.4S3, C4.2D45, C20.2D9, C36.2D5, C18.Dic5, C2.Dic45, C180.2C2, C12.4D15, C30.5Dic3, C6.1Dic15, C10.2Dic9, C5⋊2(C9⋊C8), C9⋊(C5⋊2C8), C15.2(C3⋊C8), C3.(C15⋊3C8), SmallGroup(360,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C45 — C45⋊3C8 |
Generators and relations for C45⋊3C8
G = < a,b | a45=b8=1, bab-1=a-1 >
Smallest permutation representation of C45⋊3C8
►Regular action on 360 points
Generators in S360
(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)
(1 316 155 257 56 285 135 221)(2 360 156 256 57 284 91 220)(3 359 157 255 58 283 92 219)(4 358 158 254 59 282 93 218)(5 357 159 253 60 281 94 217)(6 356 160 252 61 280 95 216)(7 355 161 251 62 279 96 215)(8 354 162 250 63 278 97 214)(9 353 163 249 64 277 98 213)(10 352 164 248 65 276 99 212)(11 351 165 247 66 275 100 211)(12 350 166 246 67 274 101 210)(13 349 167 245 68 273 102 209)(14 348 168 244 69 272 103 208)(15 347 169 243 70 271 104 207)(16 346 170 242 71 315 105 206)(17 345 171 241 72 314 106 205)(18 344 172 240 73 313 107 204)(19 343 173 239 74 312 108 203)(20 342 174 238 75 311 109 202)(21 341 175 237 76 310 110 201)(22 340 176 236 77 309 111 200)(23 339 177 235 78 308 112 199)(24 338 178 234 79 307 113 198)(25 337 179 233 80 306 114 197)(26 336 180 232 81 305 115 196)(27 335 136 231 82 304 116 195)(28 334 137 230 83 303 117 194)(29 333 138 229 84 302 118 193)(30 332 139 228 85 301 119 192)(31 331 140 227 86 300 120 191)(32 330 141 226 87 299 121 190)(33 329 142 270 88 298 122 189)(34 328 143 269 89 297 123 188)(35 327 144 268 90 296 124 187)(36 326 145 267 46 295 125 186)(37 325 146 266 47 294 126 185)(38 324 147 265 48 293 127 184)(39 323 148 264 49 292 128 183)(40 322 149 263 50 291 129 182)(41 321 150 262 51 290 130 181)(42 320 151 261 52 289 131 225)(43 319 152 260 53 288 132 224)(44 318 153 259 54 287 133 223)(45 317 154 258 55 286 134 222)
G:=sub<Sym(360)| (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), (1,316,155,257,56,285,135,221)(2,360,156,256,57,284,91,220)(3,359,157,255,58,283,92,219)(4,358,158,254,59,282,93,218)(5,357,159,253,60,281,94,217)(6,356,160,252,61,280,95,216)(7,355,161,251,62,279,96,215)(8,354,162,250,63,278,97,214)(9,353,163,249,64,277,98,213)(10,352,164,248,65,276,99,212)(11,351,165,247,66,275,100,211)(12,350,166,246,67,274,101,210)(13,349,167,245,68,273,102,209)(14,348,168,244,69,272,103,208)(15,347,169,243,70,271,104,207)(16,346,170,242,71,315,105,206)(17,345,171,241,72,314,106,205)(18,344,172,240,73,313,107,204)(19,343,173,239,74,312,108,203)(20,342,174,238,75,311,109,202)(21,341,175,237,76,310,110,201)(22,340,176,236,77,309,111,200)(23,339,177,235,78,308,112,199)(24,338,178,234,79,307,113,198)(25,337,179,233,80,306,114,197)(26,336,180,232,81,305,115,196)(27,335,136,231,82,304,116,195)(28,334,137,230,83,303,117,194)(29,333,138,229,84,302,118,193)(30,332,139,228,85,301,119,192)(31,331,140,227,86,300,120,191)(32,330,141,226,87,299,121,190)(33,329,142,270,88,298,122,189)(34,328,143,269,89,297,123,188)(35,327,144,268,90,296,124,187)(36,326,145,267,46,295,125,186)(37,325,146,266,47,294,126,185)(38,324,147,265,48,293,127,184)(39,323,148,264,49,292,128,183)(40,322,149,263,50,291,129,182)(41,321,150,262,51,290,130,181)(42,320,151,261,52,289,131,225)(43,319,152,260,53,288,132,224)(44,318,153,259,54,287,133,223)(45,317,154,258,55,286,134,222)>;
G:=Group( (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), (1,316,155,257,56,285,135,221)(2,360,156,256,57,284,91,220)(3,359,157,255,58,283,92,219)(4,358,158,254,59,282,93,218)(5,357,159,253,60,281,94,217)(6,356,160,252,61,280,95,216)(7,355,161,251,62,279,96,215)(8,354,162,250,63,278,97,214)(9,353,163,249,64,277,98,213)(10,352,164,248,65,276,99,212)(11,351,165,247,66,275,100,211)(12,350,166,246,67,274,101,210)(13,349,167,245,68,273,102,209)(14,348,168,244,69,272,103,208)(15,347,169,243,70,271,104,207)(16,346,170,242,71,315,105,206)(17,345,171,241,72,314,106,205)(18,344,172,240,73,313,107,204)(19,343,173,239,74,312,108,203)(20,342,174,238,75,311,109,202)(21,341,175,237,76,310,110,201)(22,340,176,236,77,309,111,200)(23,339,177,235,78,308,112,199)(24,338,178,234,79,307,113,198)(25,337,179,233,80,306,114,197)(26,336,180,232,81,305,115,196)(27,335,136,231,82,304,116,195)(28,334,137,230,83,303,117,194)(29,333,138,229,84,302,118,193)(30,332,139,228,85,301,119,192)(31,331,140,227,86,300,120,191)(32,330,141,226,87,299,121,190)(33,329,142,270,88,298,122,189)(34,328,143,269,89,297,123,188)(35,327,144,268,90,296,124,187)(36,326,145,267,46,295,125,186)(37,325,146,266,47,294,126,185)(38,324,147,265,48,293,127,184)(39,323,148,264,49,292,128,183)(40,322,149,263,50,291,129,182)(41,321,150,262,51,290,130,181)(42,320,151,261,52,289,131,225)(43,319,152,260,53,288,132,224)(44,318,153,259,54,287,133,223)(45,317,154,258,55,286,134,222) );
G=PermutationGroup([[(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)], [(1,316,155,257,56,285,135,221),(2,360,156,256,57,284,91,220),(3,359,157,255,58,283,92,219),(4,358,158,254,59,282,93,218),(5,357,159,253,60,281,94,217),(6,356,160,252,61,280,95,216),(7,355,161,251,62,279,96,215),(8,354,162,250,63,278,97,214),(9,353,163,249,64,277,98,213),(10,352,164,248,65,276,99,212),(11,351,165,247,66,275,100,211),(12,350,166,246,67,274,101,210),(13,349,167,245,68,273,102,209),(14,348,168,244,69,272,103,208),(15,347,169,243,70,271,104,207),(16,346,170,242,71,315,105,206),(17,345,171,241,72,314,106,205),(18,344,172,240,73,313,107,204),(19,343,173,239,74,312,108,203),(20,342,174,238,75,311,109,202),(21,341,175,237,76,310,110,201),(22,340,176,236,77,309,111,200),(23,339,177,235,78,308,112,199),(24,338,178,234,79,307,113,198),(25,337,179,233,80,306,114,197),(26,336,180,232,81,305,115,196),(27,335,136,231,82,304,116,195),(28,334,137,230,83,303,117,194),(29,333,138,229,84,302,118,193),(30,332,139,228,85,301,119,192),(31,331,140,227,86,300,120,191),(32,330,141,226,87,299,121,190),(33,329,142,270,88,298,122,189),(34,328,143,269,89,297,123,188),(35,327,144,268,90,296,124,187),(36,326,145,267,46,295,125,186),(37,325,146,266,47,294,126,185),(38,324,147,265,48,293,127,184),(39,323,148,264,49,292,128,183),(40,322,149,263,50,291,129,182),(41,321,150,262,51,290,130,181),(42,320,151,261,52,289,131,225),(43,319,152,260,53,288,132,224),(44,318,153,259,54,287,133,223),(45,317,154,258,55,286,134,222)]])
96 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 5A | 5B | 6 | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 10A | 10B | 12A | 12B | 15A | 15B | 15C | 15D | 18A | 18B | 18C | 20A | 20B | 20C | 20D | 30A | 30B | 30C | 30D | 36A | ··· | 36F | 45A | ··· | 45L | 60A | ··· | 60H | 90A | ··· | 90L | 180A | ··· | 180X |
| order | 1 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 18 | 18 | 18 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 36 | ··· | 36 | 45 | ··· | 45 | 60 | ··· | 60 | 90 | ··· | 90 | 180 | ··· | 180 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 45 | 45 | 45 | 45 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | - | + | - | |||||||
| image | C1 | C2 | C4 | C8 | S3 | D5 | Dic3 | D9 | Dic5 | C3⋊C8 | D15 | Dic9 | C5⋊2C8 | Dic15 | C9⋊C8 | D45 | C15⋊3C8 | Dic45 | C45⋊3C8 |
| kernel | C45⋊3C8 | C180 | C90 | C45 | C60 | C36 | C30 | C20 | C18 | C15 | C12 | C10 | C9 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 1 | 2 | 1 | 3 | 2 | 2 | 4 | 3 | 4 | 4 | 6 | 12 | 8 | 12 | 24 |
Matrix representation of C45⋊3C8 ►in GL2(𝔽1801) generated by
| 330 | 920 |
| 881 | 1250 |
| 137 | 940 |
| 1077 | 1664 |
G:=sub<GL(2,GF(1801))| [330,881,920,1250],[137,1077,940,1664] >;
C45⋊3C8 in GAP, Magma, Sage, TeX
C_{45}\rtimes_3C_8 % in TeX
G:=Group("C45:3C8"); // GroupNames label
G:=SmallGroup(360,3);
// by ID
G=gap.SmallGroup(360,3);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-5,-3,12,31,3267,741,2884,8645]);
// Polycyclic
G:=Group<a,b|a^45=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
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