direct product, abelian, monomial, 3-elementary
Aliases: C9×C36, SmallGroup(324,26)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C9×C36 |
| C1 — C9×C36 |
| C1 — C9×C36 |
Generators and relations for C9×C36
G = < a,b | a9=b36=1, ab=ba >
Subgroups: 69, all normal (9 characteristic)
C1, C2, C3, C4, C6, C9, C32, C12, C18, C3×C6, C3×C9, C36, C3×C12, C3×C18, C92, C3×C36, C9×C18, C9×C36
Quotients: C1, C2, C3, C4, C6, C9, C32, C12, C18, C3×C6, C3×C9, C36, C3×C12, C3×C18, C92, C3×C36, C9×C18, C9×C36
►Regular action on 324 points
Generators in S324
(1 214 295 286 179 240 84 116 72)(2 215 296 287 180 241 85 117 37)(3 216 297 288 145 242 86 118 38)(4 181 298 253 146 243 87 119 39)(5 182 299 254 147 244 88 120 40)(6 183 300 255 148 245 89 121 41)(7 184 301 256 149 246 90 122 42)(8 185 302 257 150 247 91 123 43)(9 186 303 258 151 248 92 124 44)(10 187 304 259 152 249 93 125 45)(11 188 305 260 153 250 94 126 46)(12 189 306 261 154 251 95 127 47)(13 190 307 262 155 252 96 128 48)(14 191 308 263 156 217 97 129 49)(15 192 309 264 157 218 98 130 50)(16 193 310 265 158 219 99 131 51)(17 194 311 266 159 220 100 132 52)(18 195 312 267 160 221 101 133 53)(19 196 313 268 161 222 102 134 54)(20 197 314 269 162 223 103 135 55)(21 198 315 270 163 224 104 136 56)(22 199 316 271 164 225 105 137 57)(23 200 317 272 165 226 106 138 58)(24 201 318 273 166 227 107 139 59)(25 202 319 274 167 228 108 140 60)(26 203 320 275 168 229 73 141 61)(27 204 321 276 169 230 74 142 62)(28 205 322 277 170 231 75 143 63)(29 206 323 278 171 232 76 144 64)(30 207 324 279 172 233 77 109 65)(31 208 289 280 173 234 78 110 66)(32 209 290 281 174 235 79 111 67)(33 210 291 282 175 236 80 112 68)(34 211 292 283 176 237 81 113 69)(35 212 293 284 177 238 82 114 70)(36 213 294 285 178 239 83 115 71)
(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)
G:=sub<Sym(324)| (1,214,295,286,179,240,84,116,72)(2,215,296,287,180,241,85,117,37)(3,216,297,288,145,242,86,118,38)(4,181,298,253,146,243,87,119,39)(5,182,299,254,147,244,88,120,40)(6,183,300,255,148,245,89,121,41)(7,184,301,256,149,246,90,122,42)(8,185,302,257,150,247,91,123,43)(9,186,303,258,151,248,92,124,44)(10,187,304,259,152,249,93,125,45)(11,188,305,260,153,250,94,126,46)(12,189,306,261,154,251,95,127,47)(13,190,307,262,155,252,96,128,48)(14,191,308,263,156,217,97,129,49)(15,192,309,264,157,218,98,130,50)(16,193,310,265,158,219,99,131,51)(17,194,311,266,159,220,100,132,52)(18,195,312,267,160,221,101,133,53)(19,196,313,268,161,222,102,134,54)(20,197,314,269,162,223,103,135,55)(21,198,315,270,163,224,104,136,56)(22,199,316,271,164,225,105,137,57)(23,200,317,272,165,226,106,138,58)(24,201,318,273,166,227,107,139,59)(25,202,319,274,167,228,108,140,60)(26,203,320,275,168,229,73,141,61)(27,204,321,276,169,230,74,142,62)(28,205,322,277,170,231,75,143,63)(29,206,323,278,171,232,76,144,64)(30,207,324,279,172,233,77,109,65)(31,208,289,280,173,234,78,110,66)(32,209,290,281,174,235,79,111,67)(33,210,291,282,175,236,80,112,68)(34,211,292,283,176,237,81,113,69)(35,212,293,284,177,238,82,114,70)(36,213,294,285,178,239,83,115,71), (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)>;
G:=Group( (1,214,295,286,179,240,84,116,72)(2,215,296,287,180,241,85,117,37)(3,216,297,288,145,242,86,118,38)(4,181,298,253,146,243,87,119,39)(5,182,299,254,147,244,88,120,40)(6,183,300,255,148,245,89,121,41)(7,184,301,256,149,246,90,122,42)(8,185,302,257,150,247,91,123,43)(9,186,303,258,151,248,92,124,44)(10,187,304,259,152,249,93,125,45)(11,188,305,260,153,250,94,126,46)(12,189,306,261,154,251,95,127,47)(13,190,307,262,155,252,96,128,48)(14,191,308,263,156,217,97,129,49)(15,192,309,264,157,218,98,130,50)(16,193,310,265,158,219,99,131,51)(17,194,311,266,159,220,100,132,52)(18,195,312,267,160,221,101,133,53)(19,196,313,268,161,222,102,134,54)(20,197,314,269,162,223,103,135,55)(21,198,315,270,163,224,104,136,56)(22,199,316,271,164,225,105,137,57)(23,200,317,272,165,226,106,138,58)(24,201,318,273,166,227,107,139,59)(25,202,319,274,167,228,108,140,60)(26,203,320,275,168,229,73,141,61)(27,204,321,276,169,230,74,142,62)(28,205,322,277,170,231,75,143,63)(29,206,323,278,171,232,76,144,64)(30,207,324,279,172,233,77,109,65)(31,208,289,280,173,234,78,110,66)(32,209,290,281,174,235,79,111,67)(33,210,291,282,175,236,80,112,68)(34,211,292,283,176,237,81,113,69)(35,212,293,284,177,238,82,114,70)(36,213,294,285,178,239,83,115,71), (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) );
G=PermutationGroup([[(1,214,295,286,179,240,84,116,72),(2,215,296,287,180,241,85,117,37),(3,216,297,288,145,242,86,118,38),(4,181,298,253,146,243,87,119,39),(5,182,299,254,147,244,88,120,40),(6,183,300,255,148,245,89,121,41),(7,184,301,256,149,246,90,122,42),(8,185,302,257,150,247,91,123,43),(9,186,303,258,151,248,92,124,44),(10,187,304,259,152,249,93,125,45),(11,188,305,260,153,250,94,126,46),(12,189,306,261,154,251,95,127,47),(13,190,307,262,155,252,96,128,48),(14,191,308,263,156,217,97,129,49),(15,192,309,264,157,218,98,130,50),(16,193,310,265,158,219,99,131,51),(17,194,311,266,159,220,100,132,52),(18,195,312,267,160,221,101,133,53),(19,196,313,268,161,222,102,134,54),(20,197,314,269,162,223,103,135,55),(21,198,315,270,163,224,104,136,56),(22,199,316,271,164,225,105,137,57),(23,200,317,272,165,226,106,138,58),(24,201,318,273,166,227,107,139,59),(25,202,319,274,167,228,108,140,60),(26,203,320,275,168,229,73,141,61),(27,204,321,276,169,230,74,142,62),(28,205,322,277,170,231,75,143,63),(29,206,323,278,171,232,76,144,64),(30,207,324,279,172,233,77,109,65),(31,208,289,280,173,234,78,110,66),(32,209,290,281,174,235,79,111,67),(33,210,291,282,175,236,80,112,68),(34,211,292,283,176,237,81,113,69),(35,212,293,284,177,238,82,114,70),(36,213,294,285,178,239,83,115,71)], [(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)]])
324 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 4A | 4B | 6A | ··· | 6H | 9A | ··· | 9BT | 12A | ··· | 12P | 18A | ··· | 18BT | 36A | ··· | 36EN |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
324 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | |||||||
| image | C1 | C2 | C3 | C4 | C6 | C9 | C12 | C18 | C36 |
| kernel | C9×C36 | C9×C18 | C3×C36 | C92 | C3×C18 | C36 | C3×C9 | C18 | C9 |
| # reps | 1 | 1 | 8 | 2 | 8 | 72 | 16 | 72 | 144 |
Matrix representation of C9×C36 ►in GL2(𝔽37) generated by
| 34 | 0 |
| 0 | 9 |
| 3 | 0 |
| 0 | 19 |
G:=sub<GL(2,GF(37))| [34,0,0,9],[3,0,0,19] >;
C9×C36 in GAP, Magma, Sage, TeX
C_9\times C_{36} % in TeX
G:=Group("C9xC36"); // GroupNames label
G:=SmallGroup(324,26);
// by ID
G=gap.SmallGroup(324,26);
# by ID
G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,223,338]);
// Polycyclic
G:=Group<a,b|a^9=b^36=1,a*b=b*a>;
// generators/relations