Copied to
clipboard

## G = C182order 324 = 22·34

### Abelian group of type [18,18]

Aliases: C182, SmallGroup(324,81)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C182
 Chief series C1 — C3 — C32 — C3×C9 — C92 — C9×C18 — C182
 Lower central C1 — C182
 Upper central C1 — C182

Generators and relations for C182
G = < a,b | a18=b18=1, ab=ba >

Subgroups: 115, all normal (6 characteristic)
C1, C2 [×3], C3 [×4], C22, C6 [×12], C9 [×12], C32, C2×C6 [×4], C18 [×36], C3×C6 [×3], C3×C9 [×4], C2×C18 [×12], C62, C3×C18 [×12], C92, C6×C18 [×4], C9×C18 [×3], C182
Quotients: C1, C2 [×3], C3 [×4], C22, C6 [×12], C9 [×12], C32, C2×C6 [×4], C18 [×36], C3×C6 [×3], C3×C9 [×4], C2×C18 [×12], C62, C3×C18 [×12], C92, C6×C18 [×4], C9×C18 [×3], C182

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

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

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

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

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`

׿
×
𝔽