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G = C192order 361 = 192

Elementary abelian group of type [19,19]

direct product, p-group, elementary abelian, monomial

Aliases: C192, SmallGroup(361,2)

Series: Derived Chief Lower central Upper central Jennings

C1 — C192
C1C19 — C192
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 99 307 26 345 262 201 267 332 125 48 156 287 244 188 147 82 212 74)(2 100 308 27 346 263 202 268 333 126 49 157 288 245 189 148 83 213 75)(3 101 309 28 347 264 203 269 334 127 50 158 289 246 190 149 84 214 76)(4 102 310 29 348 265 204 270 335 128 51 159 290 247 172 150 85 215 58)(5 103 311 30 349 266 205 271 336 129 52 160 291 229 173 151 86 216 59)(6 104 312 31 350 248 206 272 337 130 53 161 292 230 174 152 87 217 60)(7 105 313 32 351 249 207 273 338 131 54 162 293 231 175 134 88 218 61)(8 106 314 33 352 250 208 274 339 132 55 163 294 232 176 135 89 219 62)(9 107 315 34 353 251 209 275 340 133 56 164 295 233 177 136 90 220 63)(10 108 316 35 354 252 191 276 341 115 57 165 296 234 178 137 91 221 64)(11 109 317 36 355 253 192 277 342 116 39 166 297 235 179 138 92 222 65)(12 110 318 37 356 254 193 278 324 117 40 167 298 236 180 139 93 223 66)(13 111 319 38 357 255 194 279 325 118 41 168 299 237 181 140 94 224 67)(14 112 320 20 358 256 195 280 326 119 42 169 300 238 182 141 95 225 68)(15 113 321 21 359 257 196 281 327 120 43 170 301 239 183 142 77 226 69)(16 114 322 22 360 258 197 282 328 121 44 171 302 240 184 143 78 227 70)(17 96 323 23 361 259 198 283 329 122 45 153 303 241 185 144 79 228 71)(18 97 305 24 343 260 199 284 330 123 46 154 304 242 186 145 80 210 72)(19 98 306 25 344 261 200 285 331 124 47 155 286 243 187 146 81 211 73)

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), (1,99,307,26,345,262,201,267,332,125,48,156,287,244,188,147,82,212,74)(2,100,308,27,346,263,202,268,333,126,49,157,288,245,189,148,83,213,75)(3,101,309,28,347,264,203,269,334,127,50,158,289,246,190,149,84,214,76)(4,102,310,29,348,265,204,270,335,128,51,159,290,247,172,150,85,215,58)(5,103,311,30,349,266,205,271,336,129,52,160,291,229,173,151,86,216,59)(6,104,312,31,350,248,206,272,337,130,53,161,292,230,174,152,87,217,60)(7,105,313,32,351,249,207,273,338,131,54,162,293,231,175,134,88,218,61)(8,106,314,33,352,250,208,274,339,132,55,163,294,232,176,135,89,219,62)(9,107,315,34,353,251,209,275,340,133,56,164,295,233,177,136,90,220,63)(10,108,316,35,354,252,191,276,341,115,57,165,296,234,178,137,91,221,64)(11,109,317,36,355,253,192,277,342,116,39,166,297,235,179,138,92,222,65)(12,110,318,37,356,254,193,278,324,117,40,167,298,236,180,139,93,223,66)(13,111,319,38,357,255,194,279,325,118,41,168,299,237,181,140,94,224,67)(14,112,320,20,358,256,195,280,326,119,42,169,300,238,182,141,95,225,68)(15,113,321,21,359,257,196,281,327,120,43,170,301,239,183,142,77,226,69)(16,114,322,22,360,258,197,282,328,121,44,171,302,240,184,143,78,227,70)(17,96,323,23,361,259,198,283,329,122,45,153,303,241,185,144,79,228,71)(18,97,305,24,343,260,199,284,330,123,46,154,304,242,186,145,80,210,72)(19,98,306,25,344,261,200,285,331,124,47,155,286,243,187,146,81,211,73)>;

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), (1,99,307,26,345,262,201,267,332,125,48,156,287,244,188,147,82,212,74)(2,100,308,27,346,263,202,268,333,126,49,157,288,245,189,148,83,213,75)(3,101,309,28,347,264,203,269,334,127,50,158,289,246,190,149,84,214,76)(4,102,310,29,348,265,204,270,335,128,51,159,290,247,172,150,85,215,58)(5,103,311,30,349,266,205,271,336,129,52,160,291,229,173,151,86,216,59)(6,104,312,31,350,248,206,272,337,130,53,161,292,230,174,152,87,217,60)(7,105,313,32,351,249,207,273,338,131,54,162,293,231,175,134,88,218,61)(8,106,314,33,352,250,208,274,339,132,55,163,294,232,176,135,89,219,62)(9,107,315,34,353,251,209,275,340,133,56,164,295,233,177,136,90,220,63)(10,108,316,35,354,252,191,276,341,115,57,165,296,234,178,137,91,221,64)(11,109,317,36,355,253,192,277,342,116,39,166,297,235,179,138,92,222,65)(12,110,318,37,356,254,193,278,324,117,40,167,298,236,180,139,93,223,66)(13,111,319,38,357,255,194,279,325,118,41,168,299,237,181,140,94,224,67)(14,112,320,20,358,256,195,280,326,119,42,169,300,238,182,141,95,225,68)(15,113,321,21,359,257,196,281,327,120,43,170,301,239,183,142,77,226,69)(16,114,322,22,360,258,197,282,328,121,44,171,302,240,184,143,78,227,70)(17,96,323,23,361,259,198,283,329,122,45,153,303,241,185,144,79,228,71)(18,97,305,24,343,260,199,284,330,123,46,154,304,242,186,145,80,210,72)(19,98,306,25,344,261,200,285,331,124,47,155,286,243,187,146,81,211,73) );

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)], [(1,99,307,26,345,262,201,267,332,125,48,156,287,244,188,147,82,212,74),(2,100,308,27,346,263,202,268,333,126,49,157,288,245,189,148,83,213,75),(3,101,309,28,347,264,203,269,334,127,50,158,289,246,190,149,84,214,76),(4,102,310,29,348,265,204,270,335,128,51,159,290,247,172,150,85,215,58),(5,103,311,30,349,266,205,271,336,129,52,160,291,229,173,151,86,216,59),(6,104,312,31,350,248,206,272,337,130,53,161,292,230,174,152,87,217,60),(7,105,313,32,351,249,207,273,338,131,54,162,293,231,175,134,88,218,61),(8,106,314,33,352,250,208,274,339,132,55,163,294,232,176,135,89,219,62),(9,107,315,34,353,251,209,275,340,133,56,164,295,233,177,136,90,220,63),(10,108,316,35,354,252,191,276,341,115,57,165,296,234,178,137,91,221,64),(11,109,317,36,355,253,192,277,342,116,39,166,297,235,179,138,92,222,65),(12,110,318,37,356,254,193,278,324,117,40,167,298,236,180,139,93,223,66),(13,111,319,38,357,255,194,279,325,118,41,168,299,237,181,140,94,224,67),(14,112,320,20,358,256,195,280,326,119,42,169,300,238,182,141,95,225,68),(15,113,321,21,359,257,196,281,327,120,43,170,301,239,183,142,77,226,69),(16,114,322,22,360,258,197,282,328,121,44,171,302,240,184,143,78,227,70),(17,96,323,23,361,259,198,283,329,122,45,153,303,241,185,144,79,228,71),(18,97,305,24,343,260,199,284,330,123,46,154,304,242,186,145,80,210,72),(19,98,306,25,344,261,200,285,331,124,47,155,286,243,187,146,81,211,73)])

361 conjugacy classes

class 1 19A···19MV
order119···19
size11···1

361 irreducible representations

dim11
type+
imageC1C19
kernelC192C19
# reps1360

Matrix representation of C192 in GL2(𝔽191) generated by

1070
069
,
1600
030
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

Export

Subgroup lattice of C192 in TeX

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