Copied to
clipboard

G = C3×C111order 333 = 32·37

Abelian group of type [3,111]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C111, SmallGroup(333,5)

Series: Derived Chief Lower central Upper central

C1 — C3×C111
C1C37C111 — C3×C111
C1 — C3×C111
C1 — C3×C111

Generators and relations for C3×C111
 G = < a,b | a3=b111=1, ab=ba >


Smallest permutation representation of C3×C111
Regular action on 333 points
Generators in S333
(1 240 128)(2 241 129)(3 242 130)(4 243 131)(5 244 132)(6 245 133)(7 246 134)(8 247 135)(9 248 136)(10 249 137)(11 250 138)(12 251 139)(13 252 140)(14 253 141)(15 254 142)(16 255 143)(17 256 144)(18 257 145)(19 258 146)(20 259 147)(21 260 148)(22 261 149)(23 262 150)(24 263 151)(25 264 152)(26 265 153)(27 266 154)(28 267 155)(29 268 156)(30 269 157)(31 270 158)(32 271 159)(33 272 160)(34 273 161)(35 274 162)(36 275 163)(37 276 164)(38 277 165)(39 278 166)(40 279 167)(41 280 168)(42 281 169)(43 282 170)(44 283 171)(45 284 172)(46 285 173)(47 286 174)(48 287 175)(49 288 176)(50 289 177)(51 290 178)(52 291 179)(53 292 180)(54 293 181)(55 294 182)(56 295 183)(57 296 184)(58 297 185)(59 298 186)(60 299 187)(61 300 188)(62 301 189)(63 302 190)(64 303 191)(65 304 192)(66 305 193)(67 306 194)(68 307 195)(69 308 196)(70 309 197)(71 310 198)(72 311 199)(73 312 200)(74 313 201)(75 314 202)(76 315 203)(77 316 204)(78 317 205)(79 318 206)(80 319 207)(81 320 208)(82 321 209)(83 322 210)(84 323 211)(85 324 212)(86 325 213)(87 326 214)(88 327 215)(89 328 216)(90 329 217)(91 330 218)(92 331 219)(93 332 220)(94 333 221)(95 223 222)(96 224 112)(97 225 113)(98 226 114)(99 227 115)(100 228 116)(101 229 117)(102 230 118)(103 231 119)(104 232 120)(105 233 121)(106 234 122)(107 235 123)(108 236 124)(109 237 125)(110 238 126)(111 239 127)
(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)(223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333)

G:=sub<Sym(333)| (1,240,128)(2,241,129)(3,242,130)(4,243,131)(5,244,132)(6,245,133)(7,246,134)(8,247,135)(9,248,136)(10,249,137)(11,250,138)(12,251,139)(13,252,140)(14,253,141)(15,254,142)(16,255,143)(17,256,144)(18,257,145)(19,258,146)(20,259,147)(21,260,148)(22,261,149)(23,262,150)(24,263,151)(25,264,152)(26,265,153)(27,266,154)(28,267,155)(29,268,156)(30,269,157)(31,270,158)(32,271,159)(33,272,160)(34,273,161)(35,274,162)(36,275,163)(37,276,164)(38,277,165)(39,278,166)(40,279,167)(41,280,168)(42,281,169)(43,282,170)(44,283,171)(45,284,172)(46,285,173)(47,286,174)(48,287,175)(49,288,176)(50,289,177)(51,290,178)(52,291,179)(53,292,180)(54,293,181)(55,294,182)(56,295,183)(57,296,184)(58,297,185)(59,298,186)(60,299,187)(61,300,188)(62,301,189)(63,302,190)(64,303,191)(65,304,192)(66,305,193)(67,306,194)(68,307,195)(69,308,196)(70,309,197)(71,310,198)(72,311,199)(73,312,200)(74,313,201)(75,314,202)(76,315,203)(77,316,204)(78,317,205)(79,318,206)(80,319,207)(81,320,208)(82,321,209)(83,322,210)(84,323,211)(85,324,212)(86,325,213)(87,326,214)(88,327,215)(89,328,216)(90,329,217)(91,330,218)(92,331,219)(93,332,220)(94,333,221)(95,223,222)(96,224,112)(97,225,113)(98,226,114)(99,227,115)(100,228,116)(101,229,117)(102,230,118)(103,231,119)(104,232,120)(105,233,121)(106,234,122)(107,235,123)(108,236,124)(109,237,125)(110,238,126)(111,239,127), (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)(223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333)>;

G:=Group( (1,240,128)(2,241,129)(3,242,130)(4,243,131)(5,244,132)(6,245,133)(7,246,134)(8,247,135)(9,248,136)(10,249,137)(11,250,138)(12,251,139)(13,252,140)(14,253,141)(15,254,142)(16,255,143)(17,256,144)(18,257,145)(19,258,146)(20,259,147)(21,260,148)(22,261,149)(23,262,150)(24,263,151)(25,264,152)(26,265,153)(27,266,154)(28,267,155)(29,268,156)(30,269,157)(31,270,158)(32,271,159)(33,272,160)(34,273,161)(35,274,162)(36,275,163)(37,276,164)(38,277,165)(39,278,166)(40,279,167)(41,280,168)(42,281,169)(43,282,170)(44,283,171)(45,284,172)(46,285,173)(47,286,174)(48,287,175)(49,288,176)(50,289,177)(51,290,178)(52,291,179)(53,292,180)(54,293,181)(55,294,182)(56,295,183)(57,296,184)(58,297,185)(59,298,186)(60,299,187)(61,300,188)(62,301,189)(63,302,190)(64,303,191)(65,304,192)(66,305,193)(67,306,194)(68,307,195)(69,308,196)(70,309,197)(71,310,198)(72,311,199)(73,312,200)(74,313,201)(75,314,202)(76,315,203)(77,316,204)(78,317,205)(79,318,206)(80,319,207)(81,320,208)(82,321,209)(83,322,210)(84,323,211)(85,324,212)(86,325,213)(87,326,214)(88,327,215)(89,328,216)(90,329,217)(91,330,218)(92,331,219)(93,332,220)(94,333,221)(95,223,222)(96,224,112)(97,225,113)(98,226,114)(99,227,115)(100,228,116)(101,229,117)(102,230,118)(103,231,119)(104,232,120)(105,233,121)(106,234,122)(107,235,123)(108,236,124)(109,237,125)(110,238,126)(111,239,127), (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)(223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333) );

G=PermutationGroup([(1,240,128),(2,241,129),(3,242,130),(4,243,131),(5,244,132),(6,245,133),(7,246,134),(8,247,135),(9,248,136),(10,249,137),(11,250,138),(12,251,139),(13,252,140),(14,253,141),(15,254,142),(16,255,143),(17,256,144),(18,257,145),(19,258,146),(20,259,147),(21,260,148),(22,261,149),(23,262,150),(24,263,151),(25,264,152),(26,265,153),(27,266,154),(28,267,155),(29,268,156),(30,269,157),(31,270,158),(32,271,159),(33,272,160),(34,273,161),(35,274,162),(36,275,163),(37,276,164),(38,277,165),(39,278,166),(40,279,167),(41,280,168),(42,281,169),(43,282,170),(44,283,171),(45,284,172),(46,285,173),(47,286,174),(48,287,175),(49,288,176),(50,289,177),(51,290,178),(52,291,179),(53,292,180),(54,293,181),(55,294,182),(56,295,183),(57,296,184),(58,297,185),(59,298,186),(60,299,187),(61,300,188),(62,301,189),(63,302,190),(64,303,191),(65,304,192),(66,305,193),(67,306,194),(68,307,195),(69,308,196),(70,309,197),(71,310,198),(72,311,199),(73,312,200),(74,313,201),(75,314,202),(76,315,203),(77,316,204),(78,317,205),(79,318,206),(80,319,207),(81,320,208),(82,321,209),(83,322,210),(84,323,211),(85,324,212),(86,325,213),(87,326,214),(88,327,215),(89,328,216),(90,329,217),(91,330,218),(92,331,219),(93,332,220),(94,333,221),(95,223,222),(96,224,112),(97,225,113),(98,226,114),(99,227,115),(100,228,116),(101,229,117),(102,230,118),(103,231,119),(104,232,120),(105,233,121),(106,234,122),(107,235,123),(108,236,124),(109,237,125),(110,238,126),(111,239,127)], [(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),(223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333)])

333 conjugacy classes

class 1 3A···3H37A···37AJ111A···111KB
order13···337···37111···111
size11···11···11···1

333 irreducible representations

dim1111
type+
imageC1C3C37C111
kernelC3×C111C111C32C3
# reps1836288

Matrix representation of C3×C111 in GL2(𝔽223) generated by

10
0183
,
1210
0115
G:=sub<GL(2,GF(223))| [1,0,0,183],[121,0,0,115] >;

C3×C111 in GAP, Magma, Sage, TeX

C_3\times C_{111}
% in TeX

G:=Group("C3xC111");
// GroupNames label

G:=SmallGroup(333,5);
// by ID

G=gap.SmallGroup(333,5);
# by ID

G:=PCGroup([3,-3,-3,-37]);
// Polycyclic

G:=Group<a,b|a^3=b^111=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C111 in TeX

׿
×
𝔽