metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C37⋊2C9, C111.C3, C3.(C37⋊C3), SmallGroup(333,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C37 — C37⋊2C9 |
Generators and relations for C37⋊2C9
G = < a,b | a37=b9=1, bab-1=a10 >
Smallest permutation representation of C37⋊2C9
►Regular action on 333 points
Generators in S333
(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)
(1 308 220 90 268 151 49 246 123)(2 297 193 91 294 161 50 235 133)(3 323 203 92 283 171 51 224 143)(4 312 213 93 272 181 52 250 116)(5 301 186 94 261 154 53 239 126)(6 327 196 95 287 164 54 228 136)(7 316 206 96 276 174 55 254 146)(8 305 216 97 265 184 56 243 119)(9 331 189 98 291 157 57 232 129)(10 320 199 99 280 167 58 258 139)(11 309 209 100 269 177 59 247 112)(12 298 219 101 295 150 60 236 122)(13 324 192 102 284 160 61 225 132)(14 313 202 103 273 170 62 251 142)(15 302 212 104 262 180 63 240 115)(16 328 222 105 288 153 64 229 125)(17 317 195 106 277 163 65 255 135)(18 306 205 107 266 173 66 244 145)(19 332 215 108 292 183 67 233 118)(20 321 188 109 281 156 68 259 128)(21 310 198 110 270 166 69 248 138)(22 299 208 111 296 176 70 237 148)(23 325 218 75 285 149 71 226 121)(24 314 191 76 274 159 72 252 131)(25 303 201 77 263 169 73 241 141)(26 329 211 78 289 179 74 230 114)(27 318 221 79 278 152 38 256 124)(28 307 194 80 267 162 39 245 134)(29 333 204 81 293 172 40 234 144)(30 322 214 82 282 182 41 223 117)(31 311 187 83 271 155 42 249 127)(32 300 197 84 260 165 43 238 137)(33 326 207 85 286 175 44 227 147)(34 315 217 86 275 185 45 253 120)(35 304 190 87 264 158 46 242 130)(36 330 200 88 290 168 47 231 140)(37 319 210 89 279 178 48 257 113)
G:=sub<Sym(333)| (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), (1,308,220,90,268,151,49,246,123)(2,297,193,91,294,161,50,235,133)(3,323,203,92,283,171,51,224,143)(4,312,213,93,272,181,52,250,116)(5,301,186,94,261,154,53,239,126)(6,327,196,95,287,164,54,228,136)(7,316,206,96,276,174,55,254,146)(8,305,216,97,265,184,56,243,119)(9,331,189,98,291,157,57,232,129)(10,320,199,99,280,167,58,258,139)(11,309,209,100,269,177,59,247,112)(12,298,219,101,295,150,60,236,122)(13,324,192,102,284,160,61,225,132)(14,313,202,103,273,170,62,251,142)(15,302,212,104,262,180,63,240,115)(16,328,222,105,288,153,64,229,125)(17,317,195,106,277,163,65,255,135)(18,306,205,107,266,173,66,244,145)(19,332,215,108,292,183,67,233,118)(20,321,188,109,281,156,68,259,128)(21,310,198,110,270,166,69,248,138)(22,299,208,111,296,176,70,237,148)(23,325,218,75,285,149,71,226,121)(24,314,191,76,274,159,72,252,131)(25,303,201,77,263,169,73,241,141)(26,329,211,78,289,179,74,230,114)(27,318,221,79,278,152,38,256,124)(28,307,194,80,267,162,39,245,134)(29,333,204,81,293,172,40,234,144)(30,322,214,82,282,182,41,223,117)(31,311,187,83,271,155,42,249,127)(32,300,197,84,260,165,43,238,137)(33,326,207,85,286,175,44,227,147)(34,315,217,86,275,185,45,253,120)(35,304,190,87,264,158,46,242,130)(36,330,200,88,290,168,47,231,140)(37,319,210,89,279,178,48,257,113)>;
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), (1,308,220,90,268,151,49,246,123)(2,297,193,91,294,161,50,235,133)(3,323,203,92,283,171,51,224,143)(4,312,213,93,272,181,52,250,116)(5,301,186,94,261,154,53,239,126)(6,327,196,95,287,164,54,228,136)(7,316,206,96,276,174,55,254,146)(8,305,216,97,265,184,56,243,119)(9,331,189,98,291,157,57,232,129)(10,320,199,99,280,167,58,258,139)(11,309,209,100,269,177,59,247,112)(12,298,219,101,295,150,60,236,122)(13,324,192,102,284,160,61,225,132)(14,313,202,103,273,170,62,251,142)(15,302,212,104,262,180,63,240,115)(16,328,222,105,288,153,64,229,125)(17,317,195,106,277,163,65,255,135)(18,306,205,107,266,173,66,244,145)(19,332,215,108,292,183,67,233,118)(20,321,188,109,281,156,68,259,128)(21,310,198,110,270,166,69,248,138)(22,299,208,111,296,176,70,237,148)(23,325,218,75,285,149,71,226,121)(24,314,191,76,274,159,72,252,131)(25,303,201,77,263,169,73,241,141)(26,329,211,78,289,179,74,230,114)(27,318,221,79,278,152,38,256,124)(28,307,194,80,267,162,39,245,134)(29,333,204,81,293,172,40,234,144)(30,322,214,82,282,182,41,223,117)(31,311,187,83,271,155,42,249,127)(32,300,197,84,260,165,43,238,137)(33,326,207,85,286,175,44,227,147)(34,315,217,86,275,185,45,253,120)(35,304,190,87,264,158,46,242,130)(36,330,200,88,290,168,47,231,140)(37,319,210,89,279,178,48,257,113) );
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)], [(1,308,220,90,268,151,49,246,123),(2,297,193,91,294,161,50,235,133),(3,323,203,92,283,171,51,224,143),(4,312,213,93,272,181,52,250,116),(5,301,186,94,261,154,53,239,126),(6,327,196,95,287,164,54,228,136),(7,316,206,96,276,174,55,254,146),(8,305,216,97,265,184,56,243,119),(9,331,189,98,291,157,57,232,129),(10,320,199,99,280,167,58,258,139),(11,309,209,100,269,177,59,247,112),(12,298,219,101,295,150,60,236,122),(13,324,192,102,284,160,61,225,132),(14,313,202,103,273,170,62,251,142),(15,302,212,104,262,180,63,240,115),(16,328,222,105,288,153,64,229,125),(17,317,195,106,277,163,65,255,135),(18,306,205,107,266,173,66,244,145),(19,332,215,108,292,183,67,233,118),(20,321,188,109,281,156,68,259,128),(21,310,198,110,270,166,69,248,138),(22,299,208,111,296,176,70,237,148),(23,325,218,75,285,149,71,226,121),(24,314,191,76,274,159,72,252,131),(25,303,201,77,263,169,73,241,141),(26,329,211,78,289,179,74,230,114),(27,318,221,79,278,152,38,256,124),(28,307,194,80,267,162,39,245,134),(29,333,204,81,293,172,40,234,144),(30,322,214,82,282,182,41,223,117),(31,311,187,83,271,155,42,249,127),(32,300,197,84,260,165,43,238,137),(33,326,207,85,286,175,44,227,147),(34,315,217,86,275,185,45,253,120),(35,304,190,87,264,158,46,242,130),(36,330,200,88,290,168,47,231,140),(37,319,210,89,279,178,48,257,113)]])
45 conjugacy classes
| class | 1 | 3A | 3B | 9A | ··· | 9F | 37A | ··· | 37L | 111A | ··· | 111X |
| order | 1 | 3 | 3 | 9 | ··· | 9 | 37 | ··· | 37 | 111 | ··· | 111 |
| size | 1 | 1 | 1 | 37 | ··· | 37 | 3 | ··· | 3 | 3 | ··· | 3 |
45 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 |
| type | + | ||||
| image | C1 | C3 | C9 | C37⋊C3 | C37⋊2C9 |
| kernel | C37⋊2C9 | C111 | C37 | C3 | C1 |
| # reps | 1 | 2 | 6 | 12 | 24 |
Matrix representation of C37⋊2C9 ►in GL4(𝔽1999) generated by
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 915 | 590 |
| 503 | 0 | 0 | 0 |
| 0 | 826 | 1298 | 923 |
| 0 | 829 | 436 | 1020 |
| 0 | 1203 | 1432 | 737 |
G:=sub<GL(4,GF(1999))| [1,0,0,0,0,0,0,1,0,1,0,915,0,0,1,590],[503,0,0,0,0,826,829,1203,0,1298,436,1432,0,923,1020,737] >;
C37⋊2C9 in GAP, Magma, Sage, TeX
C_{37}\rtimes_2C_9 % in TeX
G:=Group("C37:2C9"); // GroupNames label
G:=SmallGroup(333,1);
// by ID
G=gap.SmallGroup(333,1);
# by ID
G:=PCGroup([3,-3,-3,-37,9,2108]);
// Polycyclic
G:=Group<a,b|a^37=b^9=1,b*a*b^-1=a^10>;
// generators/relations
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