direct product, abelian, monomial, 3-elementary
Aliases: C3×C84, SmallGroup(252,25)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C84 |
| C1 — C3×C84 |
| C1 — C3×C84 |
Generators and relations for C3×C84
G = < a,b | a3=b84=1, ab=ba >
Smallest permutation representation of C3×C84
►Regular action on 252 points
Generators in S252
(1 183 89)(2 184 90)(3 185 91)(4 186 92)(5 187 93)(6 188 94)(7 189 95)(8 190 96)(9 191 97)(10 192 98)(11 193 99)(12 194 100)(13 195 101)(14 196 102)(15 197 103)(16 198 104)(17 199 105)(18 200 106)(19 201 107)(20 202 108)(21 203 109)(22 204 110)(23 205 111)(24 206 112)(25 207 113)(26 208 114)(27 209 115)(28 210 116)(29 211 117)(30 212 118)(31 213 119)(32 214 120)(33 215 121)(34 216 122)(35 217 123)(36 218 124)(37 219 125)(38 220 126)(39 221 127)(40 222 128)(41 223 129)(42 224 130)(43 225 131)(44 226 132)(45 227 133)(46 228 134)(47 229 135)(48 230 136)(49 231 137)(50 232 138)(51 233 139)(52 234 140)(53 235 141)(54 236 142)(55 237 143)(56 238 144)(57 239 145)(58 240 146)(59 241 147)(60 242 148)(61 243 149)(62 244 150)(63 245 151)(64 246 152)(65 247 153)(66 248 154)(67 249 155)(68 250 156)(69 251 157)(70 252 158)(71 169 159)(72 170 160)(73 171 161)(74 172 162)(75 173 163)(76 174 164)(77 175 165)(78 176 166)(79 177 167)(80 178 168)(81 179 85)(82 180 86)(83 181 87)(84 182 88)
(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)
G:=sub<Sym(252)| (1,183,89)(2,184,90)(3,185,91)(4,186,92)(5,187,93)(6,188,94)(7,189,95)(8,190,96)(9,191,97)(10,192,98)(11,193,99)(12,194,100)(13,195,101)(14,196,102)(15,197,103)(16,198,104)(17,199,105)(18,200,106)(19,201,107)(20,202,108)(21,203,109)(22,204,110)(23,205,111)(24,206,112)(25,207,113)(26,208,114)(27,209,115)(28,210,116)(29,211,117)(30,212,118)(31,213,119)(32,214,120)(33,215,121)(34,216,122)(35,217,123)(36,218,124)(37,219,125)(38,220,126)(39,221,127)(40,222,128)(41,223,129)(42,224,130)(43,225,131)(44,226,132)(45,227,133)(46,228,134)(47,229,135)(48,230,136)(49,231,137)(50,232,138)(51,233,139)(52,234,140)(53,235,141)(54,236,142)(55,237,143)(56,238,144)(57,239,145)(58,240,146)(59,241,147)(60,242,148)(61,243,149)(62,244,150)(63,245,151)(64,246,152)(65,247,153)(66,248,154)(67,249,155)(68,250,156)(69,251,157)(70,252,158)(71,169,159)(72,170,160)(73,171,161)(74,172,162)(75,173,163)(76,174,164)(77,175,165)(78,176,166)(79,177,167)(80,178,168)(81,179,85)(82,180,86)(83,181,87)(84,182,88), (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)>;
G:=Group( (1,183,89)(2,184,90)(3,185,91)(4,186,92)(5,187,93)(6,188,94)(7,189,95)(8,190,96)(9,191,97)(10,192,98)(11,193,99)(12,194,100)(13,195,101)(14,196,102)(15,197,103)(16,198,104)(17,199,105)(18,200,106)(19,201,107)(20,202,108)(21,203,109)(22,204,110)(23,205,111)(24,206,112)(25,207,113)(26,208,114)(27,209,115)(28,210,116)(29,211,117)(30,212,118)(31,213,119)(32,214,120)(33,215,121)(34,216,122)(35,217,123)(36,218,124)(37,219,125)(38,220,126)(39,221,127)(40,222,128)(41,223,129)(42,224,130)(43,225,131)(44,226,132)(45,227,133)(46,228,134)(47,229,135)(48,230,136)(49,231,137)(50,232,138)(51,233,139)(52,234,140)(53,235,141)(54,236,142)(55,237,143)(56,238,144)(57,239,145)(58,240,146)(59,241,147)(60,242,148)(61,243,149)(62,244,150)(63,245,151)(64,246,152)(65,247,153)(66,248,154)(67,249,155)(68,250,156)(69,251,157)(70,252,158)(71,169,159)(72,170,160)(73,171,161)(74,172,162)(75,173,163)(76,174,164)(77,175,165)(78,176,166)(79,177,167)(80,178,168)(81,179,85)(82,180,86)(83,181,87)(84,182,88), (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) );
G=PermutationGroup([[(1,183,89),(2,184,90),(3,185,91),(4,186,92),(5,187,93),(6,188,94),(7,189,95),(8,190,96),(9,191,97),(10,192,98),(11,193,99),(12,194,100),(13,195,101),(14,196,102),(15,197,103),(16,198,104),(17,199,105),(18,200,106),(19,201,107),(20,202,108),(21,203,109),(22,204,110),(23,205,111),(24,206,112),(25,207,113),(26,208,114),(27,209,115),(28,210,116),(29,211,117),(30,212,118),(31,213,119),(32,214,120),(33,215,121),(34,216,122),(35,217,123),(36,218,124),(37,219,125),(38,220,126),(39,221,127),(40,222,128),(41,223,129),(42,224,130),(43,225,131),(44,226,132),(45,227,133),(46,228,134),(47,229,135),(48,230,136),(49,231,137),(50,232,138),(51,233,139),(52,234,140),(53,235,141),(54,236,142),(55,237,143),(56,238,144),(57,239,145),(58,240,146),(59,241,147),(60,242,148),(61,243,149),(62,244,150),(63,245,151),(64,246,152),(65,247,153),(66,248,154),(67,249,155),(68,250,156),(69,251,157),(70,252,158),(71,169,159),(72,170,160),(73,171,161),(74,172,162),(75,173,163),(76,174,164),(77,175,165),(78,176,166),(79,177,167),(80,178,168),(81,179,85),(82,180,86),(83,181,87),(84,182,88)], [(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)]])
252 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 4A | 4B | 6A | ··· | 6H | 7A | ··· | 7F | 12A | ··· | 12P | 14A | ··· | 14F | 21A | ··· | 21AV | 28A | ··· | 28L | 42A | ··· | 42AV | 84A | ··· | 84CR |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 7 | ··· | 7 | 12 | ··· | 12 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
252 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C7 | C12 | C14 | C21 | C28 | C42 | C84 |
| kernel | C3×C84 | C3×C42 | C84 | C3×C21 | C42 | C3×C12 | C21 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 8 | 2 | 8 | 6 | 16 | 6 | 48 | 12 | 48 | 96 |
Matrix representation of C3×C84 ►in GL2(𝔽337) generated by
| 208 | 0 |
| 0 | 1 |
| 8 | 0 |
| 0 | 9 |
G:=sub<GL(2,GF(337))| [208,0,0,1],[8,0,0,9] >;
C3×C84 in GAP, Magma, Sage, TeX
C_3\times C_{84} % in TeX
G:=Group("C3xC84"); // GroupNames label
G:=SmallGroup(252,25);
// by ID
G=gap.SmallGroup(252,25);
# by ID
G:=PCGroup([5,-2,-3,-3,-7,-2,630]);
// Polycyclic
G:=Group<a,b|a^3=b^84=1,a*b=b*a>;
// generators/relations
Export