metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C45⋊1C8, C18.F5, C90.1C4, C10.Dic9, Dic5.2D9, C30.1Dic3, C5⋊(C9⋊C8), C9⋊(C5⋊C8), C2.(C9⋊F5), C15.(C3⋊C8), C6.1(C3⋊F5), C3.(C15⋊C8), (C9×Dic5).2C2, (C3×Dic5).7S3, SmallGroup(360,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C45 — C45⋊C8 |
Generators and relations for C45⋊C8
G = < a,b | a45=b8=1, bab-1=a17 >
Smallest permutation representation of C45⋊C8
►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)
(1 356 173 233 75 291 125 195)(2 319 147 250 76 299 99 212)(3 327 166 267 77 307 118 184)(4 335 140 239 78 315 92 201)(5 343 159 256 79 278 111 218)(6 351 178 228 80 286 130 190)(7 359 152 245 81 294 104 207)(8 322 171 262 82 302 123 224)(9 330 145 234 83 310 97 196)(10 338 164 251 84 273 116 213)(11 346 138 268 85 281 135 185)(12 354 157 240 86 289 109 202)(13 317 176 257 87 297 128 219)(14 325 150 229 88 305 102 191)(15 333 169 246 89 313 121 208)(16 341 143 263 90 276 95 225)(17 349 162 235 46 284 114 197)(18 357 136 252 47 292 133 214)(19 320 155 269 48 300 107 186)(20 328 174 241 49 308 126 203)(21 336 148 258 50 271 100 220)(22 344 167 230 51 279 119 192)(23 352 141 247 52 287 93 209)(24 360 160 264 53 295 112 181)(25 323 179 236 54 303 131 198)(26 331 153 253 55 311 105 215)(27 339 172 270 56 274 124 187)(28 347 146 242 57 282 98 204)(29 355 165 259 58 290 117 221)(30 318 139 231 59 298 91 193)(31 326 158 248 60 306 110 210)(32 334 177 265 61 314 129 182)(33 342 151 237 62 277 103 199)(34 350 170 254 63 285 122 216)(35 358 144 226 64 293 96 188)(36 321 163 243 65 301 115 205)(37 329 137 260 66 309 134 222)(38 337 156 232 67 272 108 194)(39 345 175 249 68 280 127 211)(40 353 149 266 69 288 101 183)(41 316 168 238 70 296 120 200)(42 324 142 255 71 304 94 217)(43 332 161 227 72 312 113 189)(44 340 180 244 73 275 132 206)(45 348 154 261 74 283 106 223)
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), (1,356,173,233,75,291,125,195)(2,319,147,250,76,299,99,212)(3,327,166,267,77,307,118,184)(4,335,140,239,78,315,92,201)(5,343,159,256,79,278,111,218)(6,351,178,228,80,286,130,190)(7,359,152,245,81,294,104,207)(8,322,171,262,82,302,123,224)(9,330,145,234,83,310,97,196)(10,338,164,251,84,273,116,213)(11,346,138,268,85,281,135,185)(12,354,157,240,86,289,109,202)(13,317,176,257,87,297,128,219)(14,325,150,229,88,305,102,191)(15,333,169,246,89,313,121,208)(16,341,143,263,90,276,95,225)(17,349,162,235,46,284,114,197)(18,357,136,252,47,292,133,214)(19,320,155,269,48,300,107,186)(20,328,174,241,49,308,126,203)(21,336,148,258,50,271,100,220)(22,344,167,230,51,279,119,192)(23,352,141,247,52,287,93,209)(24,360,160,264,53,295,112,181)(25,323,179,236,54,303,131,198)(26,331,153,253,55,311,105,215)(27,339,172,270,56,274,124,187)(28,347,146,242,57,282,98,204)(29,355,165,259,58,290,117,221)(30,318,139,231,59,298,91,193)(31,326,158,248,60,306,110,210)(32,334,177,265,61,314,129,182)(33,342,151,237,62,277,103,199)(34,350,170,254,63,285,122,216)(35,358,144,226,64,293,96,188)(36,321,163,243,65,301,115,205)(37,329,137,260,66,309,134,222)(38,337,156,232,67,272,108,194)(39,345,175,249,68,280,127,211)(40,353,149,266,69,288,101,183)(41,316,168,238,70,296,120,200)(42,324,142,255,71,304,94,217)(43,332,161,227,72,312,113,189)(44,340,180,244,73,275,132,206)(45,348,154,261,74,283,106,223)>;
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), (1,356,173,233,75,291,125,195)(2,319,147,250,76,299,99,212)(3,327,166,267,77,307,118,184)(4,335,140,239,78,315,92,201)(5,343,159,256,79,278,111,218)(6,351,178,228,80,286,130,190)(7,359,152,245,81,294,104,207)(8,322,171,262,82,302,123,224)(9,330,145,234,83,310,97,196)(10,338,164,251,84,273,116,213)(11,346,138,268,85,281,135,185)(12,354,157,240,86,289,109,202)(13,317,176,257,87,297,128,219)(14,325,150,229,88,305,102,191)(15,333,169,246,89,313,121,208)(16,341,143,263,90,276,95,225)(17,349,162,235,46,284,114,197)(18,357,136,252,47,292,133,214)(19,320,155,269,48,300,107,186)(20,328,174,241,49,308,126,203)(21,336,148,258,50,271,100,220)(22,344,167,230,51,279,119,192)(23,352,141,247,52,287,93,209)(24,360,160,264,53,295,112,181)(25,323,179,236,54,303,131,198)(26,331,153,253,55,311,105,215)(27,339,172,270,56,274,124,187)(28,347,146,242,57,282,98,204)(29,355,165,259,58,290,117,221)(30,318,139,231,59,298,91,193)(31,326,158,248,60,306,110,210)(32,334,177,265,61,314,129,182)(33,342,151,237,62,277,103,199)(34,350,170,254,63,285,122,216)(35,358,144,226,64,293,96,188)(36,321,163,243,65,301,115,205)(37,329,137,260,66,309,134,222)(38,337,156,232,67,272,108,194)(39,345,175,249,68,280,127,211)(40,353,149,266,69,288,101,183)(41,316,168,238,70,296,120,200)(42,324,142,255,71,304,94,217)(43,332,161,227,72,312,113,189)(44,340,180,244,73,275,132,206)(45,348,154,261,74,283,106,223) );
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)], [(1,356,173,233,75,291,125,195),(2,319,147,250,76,299,99,212),(3,327,166,267,77,307,118,184),(4,335,140,239,78,315,92,201),(5,343,159,256,79,278,111,218),(6,351,178,228,80,286,130,190),(7,359,152,245,81,294,104,207),(8,322,171,262,82,302,123,224),(9,330,145,234,83,310,97,196),(10,338,164,251,84,273,116,213),(11,346,138,268,85,281,135,185),(12,354,157,240,86,289,109,202),(13,317,176,257,87,297,128,219),(14,325,150,229,88,305,102,191),(15,333,169,246,89,313,121,208),(16,341,143,263,90,276,95,225),(17,349,162,235,46,284,114,197),(18,357,136,252,47,292,133,214),(19,320,155,269,48,300,107,186),(20,328,174,241,49,308,126,203),(21,336,148,258,50,271,100,220),(22,344,167,230,51,279,119,192),(23,352,141,247,52,287,93,209),(24,360,160,264,53,295,112,181),(25,323,179,236,54,303,131,198),(26,331,153,253,55,311,105,215),(27,339,172,270,56,274,124,187),(28,347,146,242,57,282,98,204),(29,355,165,259,58,290,117,221),(30,318,139,231,59,298,91,193),(31,326,158,248,60,306,110,210),(32,334,177,265,61,314,129,182),(33,342,151,237,62,277,103,199),(34,350,170,254,63,285,122,216),(35,358,144,226,64,293,96,188),(36,321,163,243,65,301,115,205),(37,329,137,260,66,309,134,222),(38,337,156,232,67,272,108,194),(39,345,175,249,68,280,127,211),(40,353,149,266,69,288,101,183),(41,316,168,238,70,296,120,200),(42,324,142,255,71,304,94,217),(43,332,161,227,72,312,113,189),(44,340,180,244,73,275,132,206),(45,348,154,261,74,283,106,223)]])
42 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 5 | 6 | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 10 | 12A | 12B | 15A | 15B | 18A | 18B | 18C | 30A | 30B | 36A | ··· | 36F | 45A | ··· | 45F | 90A | ··· | 90F |
| order | 1 | 2 | 3 | 4 | 4 | 5 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 10 | 12 | 12 | 15 | 15 | 18 | 18 | 18 | 30 | 30 | 36 | ··· | 36 | 45 | ··· | 45 | 90 | ··· | 90 |
| size | 1 | 1 | 2 | 5 | 5 | 4 | 2 | 45 | 45 | 45 | 45 | 2 | 2 | 2 | 4 | 10 | 10 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | + | - | + | - | ||||||||
| image | C1 | C2 | C4 | C8 | S3 | Dic3 | D9 | C3⋊C8 | Dic9 | C9⋊C8 | F5 | C5⋊C8 | C3⋊F5 | C15⋊C8 | C9⋊F5 | C45⋊C8 |
| kernel | C45⋊C8 | C9×Dic5 | C90 | C45 | C3×Dic5 | C30 | Dic5 | C15 | C10 | C5 | C18 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 1 | 1 | 3 | 2 | 3 | 6 | 1 | 1 | 2 | 2 | 6 | 6 |
Matrix representation of C45⋊C8 ►in GL8(𝔽1801)
| 1503 | 1067 | 0 | 0 | 0 | 0 | 0 | 0 |
| 734 | 436 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1800 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1800 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 1646 | 1646 | 158 |
| 0 | 0 | 0 | 0 | 1313 | 652 | 1148 | 1644 |
| 0 | 0 | 0 | 0 | 1687 | 1304 | 156 | 1148 |
| 0 | 0 | 0 | 0 | 1687 | 156 | 1304 | 652 |
| 524 | 1337 | 0 | 0 | 0 | 0 | 0 | 0 |
| 813 | 1277 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1791 | 788 | 0 | 0 | 0 | 0 |
| 0 | 0 | 778 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1722 | 1296 | 1196 | 924 |
| 0 | 0 | 0 | 0 | 136 | 370 | 1587 | 926 |
| 0 | 0 | 0 | 0 | 1229 | 584 | 1127 | 1510 |
| 0 | 0 | 0 | 0 | 87 | 1044 | 926 | 383 |
G:=sub<GL(8,GF(1801))| [1503,734,0,0,0,0,0,0,1067,436,0,0,0,0,0,0,0,0,1800,1800,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,1313,1687,1687,0,0,0,0,1646,652,1304,156,0,0,0,0,1646,1148,156,1304,0,0,0,0,158,1644,1148,652],[524,813,0,0,0,0,0,0,1337,1277,0,0,0,0,0,0,0,0,1791,778,0,0,0,0,0,0,788,10,0,0,0,0,0,0,0,0,1722,136,1229,87,0,0,0,0,1296,370,584,1044,0,0,0,0,1196,1587,1127,926,0,0,0,0,924,926,1510,383] >;
C45⋊C8 in GAP, Magma, Sage, TeX
C_{45}\rtimes C_8 % in TeX
G:=Group("C45:C8"); // GroupNames label
G:=SmallGroup(360,6);
// by ID
G=gap.SmallGroup(360,6);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-5,-3,12,31,3267,741,2164,1450,8645]);
// Polycyclic
G:=Group<a,b|a^45=b^8=1,b*a*b^-1=a^17>;
// generators/relations
Export