direct product, p-group, elementary abelian, monomial
Aliases: C192, SmallGroup(361,2)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C192 |
| C1 — C192 |
| C1 — C192 |
Generators and relations for C192
G = < a,b | a19=b19=1, ab=ba >
Smallest permutation representation of C192
►Regular action on 361 points
Generators in S361
(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)(343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361)
(1 219 257 34 79 286 331 141 305 185 168 124 269 238 359 100 41 193 60)(2 220 258 35 80 287 332 142 306 186 169 125 270 239 360 101 42 194 61)(3 221 259 36 81 288 333 143 307 187 170 126 271 240 361 102 43 195 62)(4 222 260 37 82 289 334 144 308 188 171 127 272 241 343 103 44 196 63)(5 223 261 38 83 290 335 145 309 189 153 128 273 242 344 104 45 197 64)(6 224 262 20 84 291 336 146 310 190 154 129 274 243 345 105 46 198 65)(7 225 263 21 85 292 337 147 311 172 155 130 275 244 346 106 47 199 66)(8 226 264 22 86 293 338 148 312 173 156 131 276 245 347 107 48 200 67)(9 227 265 23 87 294 339 149 313 174 157 132 277 246 348 108 49 201 68)(10 228 266 24 88 295 340 150 314 175 158 133 278 247 349 109 50 202 69)(11 210 248 25 89 296 341 151 315 176 159 115 279 229 350 110 51 203 70)(12 211 249 26 90 297 342 152 316 177 160 116 280 230 351 111 52 204 71)(13 212 250 27 91 298 324 134 317 178 161 117 281 231 352 112 53 205 72)(14 213 251 28 92 299 325 135 318 179 162 118 282 232 353 113 54 206 73)(15 214 252 29 93 300 326 136 319 180 163 119 283 233 354 114 55 207 74)(16 215 253 30 94 301 327 137 320 181 164 120 284 234 355 96 56 208 75)(17 216 254 31 95 302 328 138 321 182 165 121 285 235 356 97 57 209 76)(18 217 255 32 77 303 329 139 322 183 166 122 267 236 357 98 39 191 58)(19 218 256 33 78 304 330 140 323 184 167 123 268 237 358 99 40 192 59)
G:=sub<Sym(361)| (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)(343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361), (1,219,257,34,79,286,331,141,305,185,168,124,269,238,359,100,41,193,60)(2,220,258,35,80,287,332,142,306,186,169,125,270,239,360,101,42,194,61)(3,221,259,36,81,288,333,143,307,187,170,126,271,240,361,102,43,195,62)(4,222,260,37,82,289,334,144,308,188,171,127,272,241,343,103,44,196,63)(5,223,261,38,83,290,335,145,309,189,153,128,273,242,344,104,45,197,64)(6,224,262,20,84,291,336,146,310,190,154,129,274,243,345,105,46,198,65)(7,225,263,21,85,292,337,147,311,172,155,130,275,244,346,106,47,199,66)(8,226,264,22,86,293,338,148,312,173,156,131,276,245,347,107,48,200,67)(9,227,265,23,87,294,339,149,313,174,157,132,277,246,348,108,49,201,68)(10,228,266,24,88,295,340,150,314,175,158,133,278,247,349,109,50,202,69)(11,210,248,25,89,296,341,151,315,176,159,115,279,229,350,110,51,203,70)(12,211,249,26,90,297,342,152,316,177,160,116,280,230,351,111,52,204,71)(13,212,250,27,91,298,324,134,317,178,161,117,281,231,352,112,53,205,72)(14,213,251,28,92,299,325,135,318,179,162,118,282,232,353,113,54,206,73)(15,214,252,29,93,300,326,136,319,180,163,119,283,233,354,114,55,207,74)(16,215,253,30,94,301,327,137,320,181,164,120,284,234,355,96,56,208,75)(17,216,254,31,95,302,328,138,321,182,165,121,285,235,356,97,57,209,76)(18,217,255,32,77,303,329,139,322,183,166,122,267,236,357,98,39,191,58)(19,218,256,33,78,304,330,140,323,184,167,123,268,237,358,99,40,192,59)>;
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)(343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361), (1,219,257,34,79,286,331,141,305,185,168,124,269,238,359,100,41,193,60)(2,220,258,35,80,287,332,142,306,186,169,125,270,239,360,101,42,194,61)(3,221,259,36,81,288,333,143,307,187,170,126,271,240,361,102,43,195,62)(4,222,260,37,82,289,334,144,308,188,171,127,272,241,343,103,44,196,63)(5,223,261,38,83,290,335,145,309,189,153,128,273,242,344,104,45,197,64)(6,224,262,20,84,291,336,146,310,190,154,129,274,243,345,105,46,198,65)(7,225,263,21,85,292,337,147,311,172,155,130,275,244,346,106,47,199,66)(8,226,264,22,86,293,338,148,312,173,156,131,276,245,347,107,48,200,67)(9,227,265,23,87,294,339,149,313,174,157,132,277,246,348,108,49,201,68)(10,228,266,24,88,295,340,150,314,175,158,133,278,247,349,109,50,202,69)(11,210,248,25,89,296,341,151,315,176,159,115,279,229,350,110,51,203,70)(12,211,249,26,90,297,342,152,316,177,160,116,280,230,351,111,52,204,71)(13,212,250,27,91,298,324,134,317,178,161,117,281,231,352,112,53,205,72)(14,213,251,28,92,299,325,135,318,179,162,118,282,232,353,113,54,206,73)(15,214,252,29,93,300,326,136,319,180,163,119,283,233,354,114,55,207,74)(16,215,253,30,94,301,327,137,320,181,164,120,284,234,355,96,56,208,75)(17,216,254,31,95,302,328,138,321,182,165,121,285,235,356,97,57,209,76)(18,217,255,32,77,303,329,139,322,183,166,122,267,236,357,98,39,191,58)(19,218,256,33,78,304,330,140,323,184,167,123,268,237,358,99,40,192,59) );
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),(343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361)], [(1,219,257,34,79,286,331,141,305,185,168,124,269,238,359,100,41,193,60),(2,220,258,35,80,287,332,142,306,186,169,125,270,239,360,101,42,194,61),(3,221,259,36,81,288,333,143,307,187,170,126,271,240,361,102,43,195,62),(4,222,260,37,82,289,334,144,308,188,171,127,272,241,343,103,44,196,63),(5,223,261,38,83,290,335,145,309,189,153,128,273,242,344,104,45,197,64),(6,224,262,20,84,291,336,146,310,190,154,129,274,243,345,105,46,198,65),(7,225,263,21,85,292,337,147,311,172,155,130,275,244,346,106,47,199,66),(8,226,264,22,86,293,338,148,312,173,156,131,276,245,347,107,48,200,67),(9,227,265,23,87,294,339,149,313,174,157,132,277,246,348,108,49,201,68),(10,228,266,24,88,295,340,150,314,175,158,133,278,247,349,109,50,202,69),(11,210,248,25,89,296,341,151,315,176,159,115,279,229,350,110,51,203,70),(12,211,249,26,90,297,342,152,316,177,160,116,280,230,351,111,52,204,71),(13,212,250,27,91,298,324,134,317,178,161,117,281,231,352,112,53,205,72),(14,213,251,28,92,299,325,135,318,179,162,118,282,232,353,113,54,206,73),(15,214,252,29,93,300,326,136,319,180,163,119,283,233,354,114,55,207,74),(16,215,253,30,94,301,327,137,320,181,164,120,284,234,355,96,56,208,75),(17,216,254,31,95,302,328,138,321,182,165,121,285,235,356,97,57,209,76),(18,217,255,32,77,303,329,139,322,183,166,122,267,236,357,98,39,191,58),(19,218,256,33,78,304,330,140,323,184,167,123,268,237,358,99,40,192,59)]])
361 conjugacy classes
| class | 1 | 19A | ··· | 19MV |
| order | 1 | 19 | ··· | 19 |
| size | 1 | 1 | ··· | 1 |
361 irreducible representations
| dim | 1 | 1 |
| type | + | |
| image | C1 | C19 |
| kernel | C192 | C19 |
| # reps | 1 | 360 |
Matrix representation of C192 ►in GL2(𝔽191) generated by
| 107 | 0 |
| 0 | 69 |
| 160 | 0 |
| 0 | 30 |
G:=sub<GL(2,GF(191))| [107,0,0,69],[160,0,0,30] >;
C192 in GAP, Magma, Sage, TeX
C_{19}^2 % in TeX
G:=Group("C19^2"); // GroupNames label
G:=SmallGroup(361,2);
// by ID
G=gap.SmallGroup(361,2);
# by ID
G:=PCGroup([2,-19,19]:ExponentLimit:=1);
// Polycyclic
G:=Group<a,b|a^19=b^19=1,a*b=b*a>;
// generators/relations
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