direct product, abelian, monomial, 3-elementary
Aliases: C3×C90, SmallGroup(270,20)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C90 |
| C1 — C3×C90 |
| C1 — C3×C90 |
Generators and relations for C3×C90
G = < a,b | a3=b90=1, ab=ba >
Smallest permutation representation of C3×C90
►Regular action on 270 points
Generators in S270
(1 270 153)(2 181 154)(3 182 155)(4 183 156)(5 184 157)(6 185 158)(7 186 159)(8 187 160)(9 188 161)(10 189 162)(11 190 163)(12 191 164)(13 192 165)(14 193 166)(15 194 167)(16 195 168)(17 196 169)(18 197 170)(19 198 171)(20 199 172)(21 200 173)(22 201 174)(23 202 175)(24 203 176)(25 204 177)(26 205 178)(27 206 179)(28 207 180)(29 208 91)(30 209 92)(31 210 93)(32 211 94)(33 212 95)(34 213 96)(35 214 97)(36 215 98)(37 216 99)(38 217 100)(39 218 101)(40 219 102)(41 220 103)(42 221 104)(43 222 105)(44 223 106)(45 224 107)(46 225 108)(47 226 109)(48 227 110)(49 228 111)(50 229 112)(51 230 113)(52 231 114)(53 232 115)(54 233 116)(55 234 117)(56 235 118)(57 236 119)(58 237 120)(59 238 121)(60 239 122)(61 240 123)(62 241 124)(63 242 125)(64 243 126)(65 244 127)(66 245 128)(67 246 129)(68 247 130)(69 248 131)(70 249 132)(71 250 133)(72 251 134)(73 252 135)(74 253 136)(75 254 137)(76 255 138)(77 256 139)(78 257 140)(79 258 141)(80 259 142)(81 260 143)(82 261 144)(83 262 145)(84 263 146)(85 264 147)(86 265 148)(87 266 149)(88 267 150)(89 268 151)(90 269 152)
(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)
G:=sub<Sym(270)| (1,270,153)(2,181,154)(3,182,155)(4,183,156)(5,184,157)(6,185,158)(7,186,159)(8,187,160)(9,188,161)(10,189,162)(11,190,163)(12,191,164)(13,192,165)(14,193,166)(15,194,167)(16,195,168)(17,196,169)(18,197,170)(19,198,171)(20,199,172)(21,200,173)(22,201,174)(23,202,175)(24,203,176)(25,204,177)(26,205,178)(27,206,179)(28,207,180)(29,208,91)(30,209,92)(31,210,93)(32,211,94)(33,212,95)(34,213,96)(35,214,97)(36,215,98)(37,216,99)(38,217,100)(39,218,101)(40,219,102)(41,220,103)(42,221,104)(43,222,105)(44,223,106)(45,224,107)(46,225,108)(47,226,109)(48,227,110)(49,228,111)(50,229,112)(51,230,113)(52,231,114)(53,232,115)(54,233,116)(55,234,117)(56,235,118)(57,236,119)(58,237,120)(59,238,121)(60,239,122)(61,240,123)(62,241,124)(63,242,125)(64,243,126)(65,244,127)(66,245,128)(67,246,129)(68,247,130)(69,248,131)(70,249,132)(71,250,133)(72,251,134)(73,252,135)(74,253,136)(75,254,137)(76,255,138)(77,256,139)(78,257,140)(79,258,141)(80,259,142)(81,260,143)(82,261,144)(83,262,145)(84,263,146)(85,264,147)(86,265,148)(87,266,149)(88,267,150)(89,268,151)(90,269,152), (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)>;
G:=Group( (1,270,153)(2,181,154)(3,182,155)(4,183,156)(5,184,157)(6,185,158)(7,186,159)(8,187,160)(9,188,161)(10,189,162)(11,190,163)(12,191,164)(13,192,165)(14,193,166)(15,194,167)(16,195,168)(17,196,169)(18,197,170)(19,198,171)(20,199,172)(21,200,173)(22,201,174)(23,202,175)(24,203,176)(25,204,177)(26,205,178)(27,206,179)(28,207,180)(29,208,91)(30,209,92)(31,210,93)(32,211,94)(33,212,95)(34,213,96)(35,214,97)(36,215,98)(37,216,99)(38,217,100)(39,218,101)(40,219,102)(41,220,103)(42,221,104)(43,222,105)(44,223,106)(45,224,107)(46,225,108)(47,226,109)(48,227,110)(49,228,111)(50,229,112)(51,230,113)(52,231,114)(53,232,115)(54,233,116)(55,234,117)(56,235,118)(57,236,119)(58,237,120)(59,238,121)(60,239,122)(61,240,123)(62,241,124)(63,242,125)(64,243,126)(65,244,127)(66,245,128)(67,246,129)(68,247,130)(69,248,131)(70,249,132)(71,250,133)(72,251,134)(73,252,135)(74,253,136)(75,254,137)(76,255,138)(77,256,139)(78,257,140)(79,258,141)(80,259,142)(81,260,143)(82,261,144)(83,262,145)(84,263,146)(85,264,147)(86,265,148)(87,266,149)(88,267,150)(89,268,151)(90,269,152), (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) );
G=PermutationGroup([[(1,270,153),(2,181,154),(3,182,155),(4,183,156),(5,184,157),(6,185,158),(7,186,159),(8,187,160),(9,188,161),(10,189,162),(11,190,163),(12,191,164),(13,192,165),(14,193,166),(15,194,167),(16,195,168),(17,196,169),(18,197,170),(19,198,171),(20,199,172),(21,200,173),(22,201,174),(23,202,175),(24,203,176),(25,204,177),(26,205,178),(27,206,179),(28,207,180),(29,208,91),(30,209,92),(31,210,93),(32,211,94),(33,212,95),(34,213,96),(35,214,97),(36,215,98),(37,216,99),(38,217,100),(39,218,101),(40,219,102),(41,220,103),(42,221,104),(43,222,105),(44,223,106),(45,224,107),(46,225,108),(47,226,109),(48,227,110),(49,228,111),(50,229,112),(51,230,113),(52,231,114),(53,232,115),(54,233,116),(55,234,117),(56,235,118),(57,236,119),(58,237,120),(59,238,121),(60,239,122),(61,240,123),(62,241,124),(63,242,125),(64,243,126),(65,244,127),(66,245,128),(67,246,129),(68,247,130),(69,248,131),(70,249,132),(71,250,133),(72,251,134),(73,252,135),(74,253,136),(75,254,137),(76,255,138),(77,256,139),(78,257,140),(79,258,141),(80,259,142),(81,260,143),(82,261,144),(83,262,145),(84,263,146),(85,264,147),(86,265,148),(87,266,149),(88,267,150),(89,268,151),(90,269,152)], [(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)]])
270 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 5A | 5B | 5C | 5D | 6A | ··· | 6H | 9A | ··· | 9R | 10A | 10B | 10C | 10D | 15A | ··· | 15AF | 18A | ··· | 18R | 30A | ··· | 30AF | 45A | ··· | 45BT | 90A | ··· | 90BT |
| order | 1 | 2 | 3 | ··· | 3 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 9 | ··· | 9 | 10 | 10 | 10 | 10 | 15 | ··· | 15 | 18 | ··· | 18 | 30 | ··· | 30 | 45 | ··· | 45 | 90 | ··· | 90 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
270 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||||||
| image | C1 | C2 | C3 | C3 | C5 | C6 | C6 | C9 | C10 | C15 | C15 | C18 | C30 | C30 | C45 | C90 |
| kernel | C3×C90 | C3×C45 | C90 | C3×C30 | C3×C18 | C45 | C3×C15 | C30 | C3×C9 | C18 | C3×C6 | C15 | C9 | C32 | C6 | C3 |
| # reps | 1 | 1 | 6 | 2 | 4 | 6 | 2 | 18 | 4 | 24 | 8 | 18 | 24 | 8 | 72 | 72 |
Matrix representation of C3×C90 ►in GL2(𝔽181) generated by
| 48 | 0 |
| 0 | 132 |
| 15 | 0 |
| 0 | 37 |
G:=sub<GL(2,GF(181))| [48,0,0,132],[15,0,0,37] >;
C3×C90 in GAP, Magma, Sage, TeX
C_3\times C_{90} % in TeX
G:=Group("C3xC90"); // GroupNames label
G:=SmallGroup(270,20);
// by ID
G=gap.SmallGroup(270,20);
# by ID
G:=PCGroup([5,-2,-3,-3,-5,-3,456]);
// Polycyclic
G:=Group<a,b|a^3=b^90=1,a*b=b*a>;
// generators/relations
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