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G = C3×C87order 261 = 32·29

Abelian group of type [3,87]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C87, SmallGroup(261,2)

Series: Derived Chief Lower central Upper central

C1 — C3×C87
C1C29C87 — C3×C87
C1 — C3×C87
C1 — C3×C87

Generators and relations for C3×C87
 G = < a,b | a3=b87=1, ab=ba >


Smallest permutation representation of C3×C87
Regular action on 261 points
Generators in S261
(1 246 129)(2 247 130)(3 248 131)(4 249 132)(5 250 133)(6 251 134)(7 252 135)(8 253 136)(9 254 137)(10 255 138)(11 256 139)(12 257 140)(13 258 141)(14 259 142)(15 260 143)(16 261 144)(17 175 145)(18 176 146)(19 177 147)(20 178 148)(21 179 149)(22 180 150)(23 181 151)(24 182 152)(25 183 153)(26 184 154)(27 185 155)(28 186 156)(29 187 157)(30 188 158)(31 189 159)(32 190 160)(33 191 161)(34 192 162)(35 193 163)(36 194 164)(37 195 165)(38 196 166)(39 197 167)(40 198 168)(41 199 169)(42 200 170)(43 201 171)(44 202 172)(45 203 173)(46 204 174)(47 205 88)(48 206 89)(49 207 90)(50 208 91)(51 209 92)(52 210 93)(53 211 94)(54 212 95)(55 213 96)(56 214 97)(57 215 98)(58 216 99)(59 217 100)(60 218 101)(61 219 102)(62 220 103)(63 221 104)(64 222 105)(65 223 106)(66 224 107)(67 225 108)(68 226 109)(69 227 110)(70 228 111)(71 229 112)(72 230 113)(73 231 114)(74 232 115)(75 233 116)(76 234 117)(77 235 118)(78 236 119)(79 237 120)(80 238 121)(81 239 122)(82 240 123)(83 241 124)(84 242 125)(85 243 126)(86 244 127)(87 245 128)
(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)

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

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

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

261 conjugacy classes

class 1 3A···3H29A···29AB87A···87HP
order13···329···2987···87
size11···11···11···1

261 irreducible representations

dim1111
type+
imageC1C3C29C87
kernelC3×C87C87C32C3
# reps1828224

Matrix representation of C3×C87 in GL2(𝔽349) generated by

2260
01
,
2740
0266
G:=sub<GL(2,GF(349))| [226,0,0,1],[274,0,0,266] >;

C3×C87 in GAP, Magma, Sage, TeX

C_3\times C_{87}
% in TeX

G:=Group("C3xC87");
// GroupNames label

G:=SmallGroup(261,2);
// by ID

G=gap.SmallGroup(261,2);
# by ID

G:=PCGroup([3,-3,-3,-29]);
// Polycyclic

G:=Group<a,b|a^3=b^87=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C87 in TeX

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