direct product, abelian, monomial
Aliases: C182, SmallGroup(324,81)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C182 |
| C1 — C182 |
| C1 — C182 |
Generators and relations for C182
G = < a,b | a18=b18=1, ab=ba >
Subgroups: 115, all normal (6 characteristic)
C1, C2, C3, C22, C6, C9, C32, C2×C6, C18, C3×C6, C3×C9, C2×C18, C62, C3×C18, C92, C6×C18, C9×C18, C182
Quotients: C1, C2, C3, C22, C6, C9, C32, C2×C6, C18, C3×C6, C3×C9, C2×C18, C62, C3×C18, C92, C6×C18, C9×C18, C182
►Regular action on 324 points
Generators in S324
(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)
(1 310 244 301 165 106 150 79 42 285 224 134 200 21 113 263 70 181)(2 311 245 302 166 107 151 80 43 286 225 135 201 22 114 264 71 182)(3 312 246 303 167 108 152 81 44 287 226 136 202 23 115 265 72 183)(4 313 247 304 168 91 153 82 45 288 227 137 203 24 116 266 55 184)(5 314 248 305 169 92 154 83 46 271 228 138 204 25 117 267 56 185)(6 315 249 306 170 93 155 84 47 272 229 139 205 26 118 268 57 186)(7 316 250 289 171 94 156 85 48 273 230 140 206 27 119 269 58 187)(8 317 251 290 172 95 157 86 49 274 231 141 207 28 120 270 59 188)(9 318 252 291 173 96 158 87 50 275 232 142 208 29 121 253 60 189)(10 319 235 292 174 97 159 88 51 276 233 143 209 30 122 254 61 190)(11 320 236 293 175 98 160 89 52 277 234 144 210 31 123 255 62 191)(12 321 237 294 176 99 161 90 53 278 217 127 211 32 124 256 63 192)(13 322 238 295 177 100 162 73 54 279 218 128 212 33 125 257 64 193)(14 323 239 296 178 101 145 74 37 280 219 129 213 34 126 258 65 194)(15 324 240 297 179 102 146 75 38 281 220 130 214 35 109 259 66 195)(16 307 241 298 180 103 147 76 39 282 221 131 215 36 110 260 67 196)(17 308 242 299 163 104 148 77 40 283 222 132 216 19 111 261 68 197)(18 309 243 300 164 105 149 78 41 284 223 133 199 20 112 262 69 198)
G:=sub<Sym(324)| (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), (1,310,244,301,165,106,150,79,42,285,224,134,200,21,113,263,70,181)(2,311,245,302,166,107,151,80,43,286,225,135,201,22,114,264,71,182)(3,312,246,303,167,108,152,81,44,287,226,136,202,23,115,265,72,183)(4,313,247,304,168,91,153,82,45,288,227,137,203,24,116,266,55,184)(5,314,248,305,169,92,154,83,46,271,228,138,204,25,117,267,56,185)(6,315,249,306,170,93,155,84,47,272,229,139,205,26,118,268,57,186)(7,316,250,289,171,94,156,85,48,273,230,140,206,27,119,269,58,187)(8,317,251,290,172,95,157,86,49,274,231,141,207,28,120,270,59,188)(9,318,252,291,173,96,158,87,50,275,232,142,208,29,121,253,60,189)(10,319,235,292,174,97,159,88,51,276,233,143,209,30,122,254,61,190)(11,320,236,293,175,98,160,89,52,277,234,144,210,31,123,255,62,191)(12,321,237,294,176,99,161,90,53,278,217,127,211,32,124,256,63,192)(13,322,238,295,177,100,162,73,54,279,218,128,212,33,125,257,64,193)(14,323,239,296,178,101,145,74,37,280,219,129,213,34,126,258,65,194)(15,324,240,297,179,102,146,75,38,281,220,130,214,35,109,259,66,195)(16,307,241,298,180,103,147,76,39,282,221,131,215,36,110,260,67,196)(17,308,242,299,163,104,148,77,40,283,222,132,216,19,111,261,68,197)(18,309,243,300,164,105,149,78,41,284,223,133,199,20,112,262,69,198)>;
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), (1,310,244,301,165,106,150,79,42,285,224,134,200,21,113,263,70,181)(2,311,245,302,166,107,151,80,43,286,225,135,201,22,114,264,71,182)(3,312,246,303,167,108,152,81,44,287,226,136,202,23,115,265,72,183)(4,313,247,304,168,91,153,82,45,288,227,137,203,24,116,266,55,184)(5,314,248,305,169,92,154,83,46,271,228,138,204,25,117,267,56,185)(6,315,249,306,170,93,155,84,47,272,229,139,205,26,118,268,57,186)(7,316,250,289,171,94,156,85,48,273,230,140,206,27,119,269,58,187)(8,317,251,290,172,95,157,86,49,274,231,141,207,28,120,270,59,188)(9,318,252,291,173,96,158,87,50,275,232,142,208,29,121,253,60,189)(10,319,235,292,174,97,159,88,51,276,233,143,209,30,122,254,61,190)(11,320,236,293,175,98,160,89,52,277,234,144,210,31,123,255,62,191)(12,321,237,294,176,99,161,90,53,278,217,127,211,32,124,256,63,192)(13,322,238,295,177,100,162,73,54,279,218,128,212,33,125,257,64,193)(14,323,239,296,178,101,145,74,37,280,219,129,213,34,126,258,65,194)(15,324,240,297,179,102,146,75,38,281,220,130,214,35,109,259,66,195)(16,307,241,298,180,103,147,76,39,282,221,131,215,36,110,260,67,196)(17,308,242,299,163,104,148,77,40,283,222,132,216,19,111,261,68,197)(18,309,243,300,164,105,149,78,41,284,223,133,199,20,112,262,69,198) );
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)], [(1,310,244,301,165,106,150,79,42,285,224,134,200,21,113,263,70,181),(2,311,245,302,166,107,151,80,43,286,225,135,201,22,114,264,71,182),(3,312,246,303,167,108,152,81,44,287,226,136,202,23,115,265,72,183),(4,313,247,304,168,91,153,82,45,288,227,137,203,24,116,266,55,184),(5,314,248,305,169,92,154,83,46,271,228,138,204,25,117,267,56,185),(6,315,249,306,170,93,155,84,47,272,229,139,205,26,118,268,57,186),(7,316,250,289,171,94,156,85,48,273,230,140,206,27,119,269,58,187),(8,317,251,290,172,95,157,86,49,274,231,141,207,28,120,270,59,188),(9,318,252,291,173,96,158,87,50,275,232,142,208,29,121,253,60,189),(10,319,235,292,174,97,159,88,51,276,233,143,209,30,122,254,61,190),(11,320,236,293,175,98,160,89,52,277,234,144,210,31,123,255,62,191),(12,321,237,294,176,99,161,90,53,278,217,127,211,32,124,256,63,192),(13,322,238,295,177,100,162,73,54,279,218,128,212,33,125,257,64,193),(14,323,239,296,178,101,145,74,37,280,219,129,213,34,126,258,65,194),(15,324,240,297,179,102,146,75,38,281,220,130,214,35,109,259,66,195),(16,307,241,298,180,103,147,76,39,282,221,131,215,36,110,260,67,196),(17,308,242,299,163,104,148,77,40,283,222,132,216,19,111,261,68,197),(18,309,243,300,164,105,149,78,41,284,223,133,199,20,112,262,69,198)]])
324 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 6A | ··· | 6X | 9A | ··· | 9BT | 18A | ··· | 18HH |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
324 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||
| image | C1 | C2 | C3 | C6 | C9 | C18 |
| kernel | C182 | C9×C18 | C6×C18 | C3×C18 | C2×C18 | C18 |
| # reps | 1 | 3 | 8 | 24 | 72 | 216 |
Matrix representation of C182 ►in GL2(𝔽19) generated by
| 11 | 0 |
| 0 | 14 |
| 14 | 0 |
| 0 | 15 |
G:=sub<GL(2,GF(19))| [11,0,0,14],[14,0,0,15] >;
C182 in GAP, Magma, Sage, TeX
C_{18}^2 % in TeX
G:=Group("C18^2"); // GroupNames label
G:=SmallGroup(324,81);
// by ID
G=gap.SmallGroup(324,81);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,176,237]);
// Polycyclic
G:=Group<a,b|a^18=b^18=1,a*b=b*a>;
// generators/relations