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G = C3×C90order 270 = 2·33·5

Abelian group of type [3,90]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C90, SmallGroup(270,20)

Series: Derived Chief Lower central Upper central

C1 — C3×C90
C1C3C15C3×C15C3×C45 — 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
(1 226 147)(2 227 148)(3 228 149)(4 229 150)(5 230 151)(6 231 152)(7 232 153)(8 233 154)(9 234 155)(10 235 156)(11 236 157)(12 237 158)(13 238 159)(14 239 160)(15 240 161)(16 241 162)(17 242 163)(18 243 164)(19 244 165)(20 245 166)(21 246 167)(22 247 168)(23 248 169)(24 249 170)(25 250 171)(26 251 172)(27 252 173)(28 253 174)(29 254 175)(30 255 176)(31 256 177)(32 257 178)(33 258 179)(34 259 180)(35 260 91)(36 261 92)(37 262 93)(38 263 94)(39 264 95)(40 265 96)(41 266 97)(42 267 98)(43 268 99)(44 269 100)(45 270 101)(46 181 102)(47 182 103)(48 183 104)(49 184 105)(50 185 106)(51 186 107)(52 187 108)(53 188 109)(54 189 110)(55 190 111)(56 191 112)(57 192 113)(58 193 114)(59 194 115)(60 195 116)(61 196 117)(62 197 118)(63 198 119)(64 199 120)(65 200 121)(66 201 122)(67 202 123)(68 203 124)(69 204 125)(70 205 126)(71 206 127)(72 207 128)(73 208 129)(74 209 130)(75 210 131)(76 211 132)(77 212 133)(78 213 134)(79 214 135)(80 215 136)(81 216 137)(82 217 138)(83 218 139)(84 219 140)(85 220 141)(86 221 142)(87 222 143)(88 223 144)(89 224 145)(90 225 146)
(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)| (1,226,147)(2,227,148)(3,228,149)(4,229,150)(5,230,151)(6,231,152)(7,232,153)(8,233,154)(9,234,155)(10,235,156)(11,236,157)(12,237,158)(13,238,159)(14,239,160)(15,240,161)(16,241,162)(17,242,163)(18,243,164)(19,244,165)(20,245,166)(21,246,167)(22,247,168)(23,248,169)(24,249,170)(25,250,171)(26,251,172)(27,252,173)(28,253,174)(29,254,175)(30,255,176)(31,256,177)(32,257,178)(33,258,179)(34,259,180)(35,260,91)(36,261,92)(37,262,93)(38,263,94)(39,264,95)(40,265,96)(41,266,97)(42,267,98)(43,268,99)(44,269,100)(45,270,101)(46,181,102)(47,182,103)(48,183,104)(49,184,105)(50,185,106)(51,186,107)(52,187,108)(53,188,109)(54,189,110)(55,190,111)(56,191,112)(57,192,113)(58,193,114)(59,194,115)(60,195,116)(61,196,117)(62,197,118)(63,198,119)(64,199,120)(65,200,121)(66,201,122)(67,202,123)(68,203,124)(69,204,125)(70,205,126)(71,206,127)(72,207,128)(73,208,129)(74,209,130)(75,210,131)(76,211,132)(77,212,133)(78,213,134)(79,214,135)(80,215,136)(81,216,137)(82,217,138)(83,218,139)(84,219,140)(85,220,141)(86,221,142)(87,222,143)(88,223,144)(89,224,145)(90,225,146), (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( (1,226,147)(2,227,148)(3,228,149)(4,229,150)(5,230,151)(6,231,152)(7,232,153)(8,233,154)(9,234,155)(10,235,156)(11,236,157)(12,237,158)(13,238,159)(14,239,160)(15,240,161)(16,241,162)(17,242,163)(18,243,164)(19,244,165)(20,245,166)(21,246,167)(22,247,168)(23,248,169)(24,249,170)(25,250,171)(26,251,172)(27,252,173)(28,253,174)(29,254,175)(30,255,176)(31,256,177)(32,257,178)(33,258,179)(34,259,180)(35,260,91)(36,261,92)(37,262,93)(38,263,94)(39,264,95)(40,265,96)(41,266,97)(42,267,98)(43,268,99)(44,269,100)(45,270,101)(46,181,102)(47,182,103)(48,183,104)(49,184,105)(50,185,106)(51,186,107)(52,187,108)(53,188,109)(54,189,110)(55,190,111)(56,191,112)(57,192,113)(58,193,114)(59,194,115)(60,195,116)(61,196,117)(62,197,118)(63,198,119)(64,199,120)(65,200,121)(66,201,122)(67,202,123)(68,203,124)(69,204,125)(70,205,126)(71,206,127)(72,207,128)(73,208,129)(74,209,130)(75,210,131)(76,211,132)(77,212,133)(78,213,134)(79,214,135)(80,215,136)(81,216,137)(82,217,138)(83,218,139)(84,219,140)(85,220,141)(86,221,142)(87,222,143)(88,223,144)(89,224,145)(90,225,146), (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([(1,226,147),(2,227,148),(3,228,149),(4,229,150),(5,230,151),(6,231,152),(7,232,153),(8,233,154),(9,234,155),(10,235,156),(11,236,157),(12,237,158),(13,238,159),(14,239,160),(15,240,161),(16,241,162),(17,242,163),(18,243,164),(19,244,165),(20,245,166),(21,246,167),(22,247,168),(23,248,169),(24,249,170),(25,250,171),(26,251,172),(27,252,173),(28,253,174),(29,254,175),(30,255,176),(31,256,177),(32,257,178),(33,258,179),(34,259,180),(35,260,91),(36,261,92),(37,262,93),(38,263,94),(39,264,95),(40,265,96),(41,266,97),(42,267,98),(43,268,99),(44,269,100),(45,270,101),(46,181,102),(47,182,103),(48,183,104),(49,184,105),(50,185,106),(51,186,107),(52,187,108),(53,188,109),(54,189,110),(55,190,111),(56,191,112),(57,192,113),(58,193,114),(59,194,115),(60,195,116),(61,196,117),(62,197,118),(63,198,119),(64,199,120),(65,200,121),(66,201,122),(67,202,123),(68,203,124),(69,204,125),(70,205,126),(71,206,127),(72,207,128),(73,208,129),(74,209,130),(75,210,131),(76,211,132),(77,212,133),(78,213,134),(79,214,135),(80,215,136),(81,216,137),(82,217,138),(83,218,139),(84,219,140),(85,220,141),(86,221,142),(87,222,143),(88,223,144),(89,224,145),(90,225,146)], [(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···3H5A5B5C5D6A···6H9A···9R10A10B10C10D15A···15AF18A···18R30A···30AF45A···45BT90A···90BT
order123···355556···69···91010101015···1518···1830···3045···4590···90
size111···111111···11···111111···11···11···11···11···1

270 irreducible representations

dim1111111111111111
type++
imageC1C2C3C3C5C6C6C9C10C15C15C18C30C30C45C90
kernelC3×C90C3×C45C90C3×C30C3×C18C45C3×C15C30C3×C9C18C3×C6C15C9C32C6C3
# reps1162462184248182487272

Matrix representation of C3×C90 in GL2(𝔽181) generated by

480
0132
,
150
037
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

Export

Subgroup lattice of C3×C90 in TeX

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