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G = C3×C99order 297 = 33·11

Abelian group of type [3,99]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C99, SmallGroup(297,2)

Series: Derived Chief Lower central Upper central

C1 — C3×C99
C1C3C33C99 — C3×C99
C1 — C3×C99
C1 — C3×C99

Generators and relations for C3×C99
 G = < a,b | a3=b99=1, ab=ba >


Smallest permutation representation of C3×C99
Regular action on 297 points
Generators in S297
(1 216 135)(2 217 136)(3 218 137)(4 219 138)(5 220 139)(6 221 140)(7 222 141)(8 223 142)(9 224 143)(10 225 144)(11 226 145)(12 227 146)(13 228 147)(14 229 148)(15 230 149)(16 231 150)(17 232 151)(18 233 152)(19 234 153)(20 235 154)(21 236 155)(22 237 156)(23 238 157)(24 239 158)(25 240 159)(26 241 160)(27 242 161)(28 243 162)(29 244 163)(30 245 164)(31 246 165)(32 247 166)(33 248 167)(34 249 168)(35 250 169)(36 251 170)(37 252 171)(38 253 172)(39 254 173)(40 255 174)(41 256 175)(42 257 176)(43 258 177)(44 259 178)(45 260 179)(46 261 180)(47 262 181)(48 263 182)(49 264 183)(50 265 184)(51 266 185)(52 267 186)(53 268 187)(54 269 188)(55 270 189)(56 271 190)(57 272 191)(58 273 192)(59 274 193)(60 275 194)(61 276 195)(62 277 196)(63 278 197)(64 279 198)(65 280 100)(66 281 101)(67 282 102)(68 283 103)(69 284 104)(70 285 105)(71 286 106)(72 287 107)(73 288 108)(74 289 109)(75 290 110)(76 291 111)(77 292 112)(78 293 113)(79 294 114)(80 295 115)(81 296 116)(82 297 117)(83 199 118)(84 200 119)(85 201 120)(86 202 121)(87 203 122)(88 204 123)(89 205 124)(90 206 125)(91 207 126)(92 208 127)(93 209 128)(94 210 129)(95 211 130)(96 212 131)(97 213 132)(98 214 133)(99 215 134)
(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)

G:=sub<Sym(297)| (1,216,135)(2,217,136)(3,218,137)(4,219,138)(5,220,139)(6,221,140)(7,222,141)(8,223,142)(9,224,143)(10,225,144)(11,226,145)(12,227,146)(13,228,147)(14,229,148)(15,230,149)(16,231,150)(17,232,151)(18,233,152)(19,234,153)(20,235,154)(21,236,155)(22,237,156)(23,238,157)(24,239,158)(25,240,159)(26,241,160)(27,242,161)(28,243,162)(29,244,163)(30,245,164)(31,246,165)(32,247,166)(33,248,167)(34,249,168)(35,250,169)(36,251,170)(37,252,171)(38,253,172)(39,254,173)(40,255,174)(41,256,175)(42,257,176)(43,258,177)(44,259,178)(45,260,179)(46,261,180)(47,262,181)(48,263,182)(49,264,183)(50,265,184)(51,266,185)(52,267,186)(53,268,187)(54,269,188)(55,270,189)(56,271,190)(57,272,191)(58,273,192)(59,274,193)(60,275,194)(61,276,195)(62,277,196)(63,278,197)(64,279,198)(65,280,100)(66,281,101)(67,282,102)(68,283,103)(69,284,104)(70,285,105)(71,286,106)(72,287,107)(73,288,108)(74,289,109)(75,290,110)(76,291,111)(77,292,112)(78,293,113)(79,294,114)(80,295,115)(81,296,116)(82,297,117)(83,199,118)(84,200,119)(85,201,120)(86,202,121)(87,203,122)(88,204,123)(89,205,124)(90,206,125)(91,207,126)(92,208,127)(93,209,128)(94,210,129)(95,211,130)(96,212,131)(97,213,132)(98,214,133)(99,215,134), (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)>;

G:=Group( (1,216,135)(2,217,136)(3,218,137)(4,219,138)(5,220,139)(6,221,140)(7,222,141)(8,223,142)(9,224,143)(10,225,144)(11,226,145)(12,227,146)(13,228,147)(14,229,148)(15,230,149)(16,231,150)(17,232,151)(18,233,152)(19,234,153)(20,235,154)(21,236,155)(22,237,156)(23,238,157)(24,239,158)(25,240,159)(26,241,160)(27,242,161)(28,243,162)(29,244,163)(30,245,164)(31,246,165)(32,247,166)(33,248,167)(34,249,168)(35,250,169)(36,251,170)(37,252,171)(38,253,172)(39,254,173)(40,255,174)(41,256,175)(42,257,176)(43,258,177)(44,259,178)(45,260,179)(46,261,180)(47,262,181)(48,263,182)(49,264,183)(50,265,184)(51,266,185)(52,267,186)(53,268,187)(54,269,188)(55,270,189)(56,271,190)(57,272,191)(58,273,192)(59,274,193)(60,275,194)(61,276,195)(62,277,196)(63,278,197)(64,279,198)(65,280,100)(66,281,101)(67,282,102)(68,283,103)(69,284,104)(70,285,105)(71,286,106)(72,287,107)(73,288,108)(74,289,109)(75,290,110)(76,291,111)(77,292,112)(78,293,113)(79,294,114)(80,295,115)(81,296,116)(82,297,117)(83,199,118)(84,200,119)(85,201,120)(86,202,121)(87,203,122)(88,204,123)(89,205,124)(90,206,125)(91,207,126)(92,208,127)(93,209,128)(94,210,129)(95,211,130)(96,212,131)(97,213,132)(98,214,133)(99,215,134), (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) );

G=PermutationGroup([(1,216,135),(2,217,136),(3,218,137),(4,219,138),(5,220,139),(6,221,140),(7,222,141),(8,223,142),(9,224,143),(10,225,144),(11,226,145),(12,227,146),(13,228,147),(14,229,148),(15,230,149),(16,231,150),(17,232,151),(18,233,152),(19,234,153),(20,235,154),(21,236,155),(22,237,156),(23,238,157),(24,239,158),(25,240,159),(26,241,160),(27,242,161),(28,243,162),(29,244,163),(30,245,164),(31,246,165),(32,247,166),(33,248,167),(34,249,168),(35,250,169),(36,251,170),(37,252,171),(38,253,172),(39,254,173),(40,255,174),(41,256,175),(42,257,176),(43,258,177),(44,259,178),(45,260,179),(46,261,180),(47,262,181),(48,263,182),(49,264,183),(50,265,184),(51,266,185),(52,267,186),(53,268,187),(54,269,188),(55,270,189),(56,271,190),(57,272,191),(58,273,192),(59,274,193),(60,275,194),(61,276,195),(62,277,196),(63,278,197),(64,279,198),(65,280,100),(66,281,101),(67,282,102),(68,283,103),(69,284,104),(70,285,105),(71,286,106),(72,287,107),(73,288,108),(74,289,109),(75,290,110),(76,291,111),(77,292,112),(78,293,113),(79,294,114),(80,295,115),(81,296,116),(82,297,117),(83,199,118),(84,200,119),(85,201,120),(86,202,121),(87,203,122),(88,204,123),(89,205,124),(90,206,125),(91,207,126),(92,208,127),(93,209,128),(94,210,129),(95,211,130),(96,212,131),(97,213,132),(98,214,133),(99,215,134)], [(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)])

297 conjugacy classes

class 1 3A···3H9A···9R11A···11J33A···33CB99A···99FX
order13···39···911···1133···3399···99
size11···11···11···11···11···1

297 irreducible representations

dim11111111
type+
imageC1C3C3C9C11C33C33C99
kernelC3×C99C99C3×C33C33C3×C9C9C32C3
# reps16218106020180

Matrix representation of C3×C99 in GL2(𝔽199) generated by

1060
01
,
520
0180
G:=sub<GL(2,GF(199))| [106,0,0,1],[52,0,0,180] >;

C3×C99 in GAP, Magma, Sage, TeX

C_3\times C_{99}
% in TeX

G:=Group("C3xC99");
// GroupNames label

G:=SmallGroup(297,2);
// by ID

G=gap.SmallGroup(297,2);
# by ID

G:=PCGroup([4,-3,-3,-11,-3,396]);
// Polycyclic

G:=Group<a,b|a^3=b^99=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C99 in TeX

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