direct product, abelian, monomial, 3-elementary
Aliases: C3×C102, SmallGroup(306,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C102 |
| C1 — C3×C102 |
| C1 — C3×C102 |
Generators and relations for C3×C102
G = < a,b | a3=b102=1, ab=ba >
Smallest permutation representation of C3×C102
►Regular action on 306 points
Generators in S306
(1 224 144)(2 225 145)(3 226 146)(4 227 147)(5 228 148)(6 229 149)(7 230 150)(8 231 151)(9 232 152)(10 233 153)(11 234 154)(12 235 155)(13 236 156)(14 237 157)(15 238 158)(16 239 159)(17 240 160)(18 241 161)(19 242 162)(20 243 163)(21 244 164)(22 245 165)(23 246 166)(24 247 167)(25 248 168)(26 249 169)(27 250 170)(28 251 171)(29 252 172)(30 253 173)(31 254 174)(32 255 175)(33 256 176)(34 257 177)(35 258 178)(36 259 179)(37 260 180)(38 261 181)(39 262 182)(40 263 183)(41 264 184)(42 265 185)(43 266 186)(44 267 187)(45 268 188)(46 269 189)(47 270 190)(48 271 191)(49 272 192)(50 273 193)(51 274 194)(52 275 195)(53 276 196)(54 277 197)(55 278 198)(56 279 199)(57 280 200)(58 281 201)(59 282 202)(60 283 203)(61 284 204)(62 285 103)(63 286 104)(64 287 105)(65 288 106)(66 289 107)(67 290 108)(68 291 109)(69 292 110)(70 293 111)(71 294 112)(72 295 113)(73 296 114)(74 297 115)(75 298 116)(76 299 117)(77 300 118)(78 301 119)(79 302 120)(80 303 121)(81 304 122)(82 305 123)(83 306 124)(84 205 125)(85 206 126)(86 207 127)(87 208 128)(88 209 129)(89 210 130)(90 211 131)(91 212 132)(92 213 133)(93 214 134)(94 215 135)(95 216 136)(96 217 137)(97 218 138)(98 219 139)(99 220 140)(100 221 141)(101 222 142)(102 223 143)
(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)
G:=sub<Sym(306)| (1,224,144)(2,225,145)(3,226,146)(4,227,147)(5,228,148)(6,229,149)(7,230,150)(8,231,151)(9,232,152)(10,233,153)(11,234,154)(12,235,155)(13,236,156)(14,237,157)(15,238,158)(16,239,159)(17,240,160)(18,241,161)(19,242,162)(20,243,163)(21,244,164)(22,245,165)(23,246,166)(24,247,167)(25,248,168)(26,249,169)(27,250,170)(28,251,171)(29,252,172)(30,253,173)(31,254,174)(32,255,175)(33,256,176)(34,257,177)(35,258,178)(36,259,179)(37,260,180)(38,261,181)(39,262,182)(40,263,183)(41,264,184)(42,265,185)(43,266,186)(44,267,187)(45,268,188)(46,269,189)(47,270,190)(48,271,191)(49,272,192)(50,273,193)(51,274,194)(52,275,195)(53,276,196)(54,277,197)(55,278,198)(56,279,199)(57,280,200)(58,281,201)(59,282,202)(60,283,203)(61,284,204)(62,285,103)(63,286,104)(64,287,105)(65,288,106)(66,289,107)(67,290,108)(68,291,109)(69,292,110)(70,293,111)(71,294,112)(72,295,113)(73,296,114)(74,297,115)(75,298,116)(76,299,117)(77,300,118)(78,301,119)(79,302,120)(80,303,121)(81,304,122)(82,305,123)(83,306,124)(84,205,125)(85,206,126)(86,207,127)(87,208,128)(88,209,129)(89,210,130)(90,211,131)(91,212,132)(92,213,133)(93,214,134)(94,215,135)(95,216,136)(96,217,137)(97,218,138)(98,219,139)(99,220,140)(100,221,141)(101,222,142)(102,223,143), (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)>;
G:=Group( (1,224,144)(2,225,145)(3,226,146)(4,227,147)(5,228,148)(6,229,149)(7,230,150)(8,231,151)(9,232,152)(10,233,153)(11,234,154)(12,235,155)(13,236,156)(14,237,157)(15,238,158)(16,239,159)(17,240,160)(18,241,161)(19,242,162)(20,243,163)(21,244,164)(22,245,165)(23,246,166)(24,247,167)(25,248,168)(26,249,169)(27,250,170)(28,251,171)(29,252,172)(30,253,173)(31,254,174)(32,255,175)(33,256,176)(34,257,177)(35,258,178)(36,259,179)(37,260,180)(38,261,181)(39,262,182)(40,263,183)(41,264,184)(42,265,185)(43,266,186)(44,267,187)(45,268,188)(46,269,189)(47,270,190)(48,271,191)(49,272,192)(50,273,193)(51,274,194)(52,275,195)(53,276,196)(54,277,197)(55,278,198)(56,279,199)(57,280,200)(58,281,201)(59,282,202)(60,283,203)(61,284,204)(62,285,103)(63,286,104)(64,287,105)(65,288,106)(66,289,107)(67,290,108)(68,291,109)(69,292,110)(70,293,111)(71,294,112)(72,295,113)(73,296,114)(74,297,115)(75,298,116)(76,299,117)(77,300,118)(78,301,119)(79,302,120)(80,303,121)(81,304,122)(82,305,123)(83,306,124)(84,205,125)(85,206,126)(86,207,127)(87,208,128)(88,209,129)(89,210,130)(90,211,131)(91,212,132)(92,213,133)(93,214,134)(94,215,135)(95,216,136)(96,217,137)(97,218,138)(98,219,139)(99,220,140)(100,221,141)(101,222,142)(102,223,143), (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) );
G=PermutationGroup([[(1,224,144),(2,225,145),(3,226,146),(4,227,147),(5,228,148),(6,229,149),(7,230,150),(8,231,151),(9,232,152),(10,233,153),(11,234,154),(12,235,155),(13,236,156),(14,237,157),(15,238,158),(16,239,159),(17,240,160),(18,241,161),(19,242,162),(20,243,163),(21,244,164),(22,245,165),(23,246,166),(24,247,167),(25,248,168),(26,249,169),(27,250,170),(28,251,171),(29,252,172),(30,253,173),(31,254,174),(32,255,175),(33,256,176),(34,257,177),(35,258,178),(36,259,179),(37,260,180),(38,261,181),(39,262,182),(40,263,183),(41,264,184),(42,265,185),(43,266,186),(44,267,187),(45,268,188),(46,269,189),(47,270,190),(48,271,191),(49,272,192),(50,273,193),(51,274,194),(52,275,195),(53,276,196),(54,277,197),(55,278,198),(56,279,199),(57,280,200),(58,281,201),(59,282,202),(60,283,203),(61,284,204),(62,285,103),(63,286,104),(64,287,105),(65,288,106),(66,289,107),(67,290,108),(68,291,109),(69,292,110),(70,293,111),(71,294,112),(72,295,113),(73,296,114),(74,297,115),(75,298,116),(76,299,117),(77,300,118),(78,301,119),(79,302,120),(80,303,121),(81,304,122),(82,305,123),(83,306,124),(84,205,125),(85,206,126),(86,207,127),(87,208,128),(88,209,129),(89,210,130),(90,211,131),(91,212,132),(92,213,133),(93,214,134),(94,215,135),(95,216,136),(96,217,137),(97,218,138),(98,219,139),(99,220,140),(100,221,141),(101,222,142),(102,223,143)], [(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)]])
306 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 6A | ··· | 6H | 17A | ··· | 17P | 34A | ··· | 34P | 51A | ··· | 51DX | 102A | ··· | 102DX |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 102 | ··· | 102 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
306 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C17 | C34 | C51 | C102 |
| kernel | C3×C102 | C3×C51 | C102 | C51 | C3×C6 | C32 | C6 | C3 |
| # reps | 1 | 1 | 8 | 8 | 16 | 16 | 128 | 128 |
Matrix representation of C3×C102 ►in GL2(𝔽103) generated by
| 56 | 0 |
| 0 | 56 |
| 26 | 0 |
| 0 | 70 |
G:=sub<GL(2,GF(103))| [56,0,0,56],[26,0,0,70] >;
C3×C102 in GAP, Magma, Sage, TeX
C_3\times C_{102} % in TeX
G:=Group("C3xC102"); // GroupNames label
G:=SmallGroup(306,10);
// by ID
G=gap.SmallGroup(306,10);
# by ID
G:=PCGroup([4,-2,-3,-3,-17]);
// Polycyclic
G:=Group<a,b|a^3=b^102=1,a*b=b*a>;
// generators/relations
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