C3^2xC33 - GroupNames

G = C₃²×C₃₃ order 297 = 3³·11

Abelian group of type [3,3,33]

direct product, abelian, monomial, 3-elementary

Aliases: C₃²×C₃₃, SmallGroup(297,5)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃²×C₃₃

Chief series

C₁ — C₁₁ — C₃₃ — C₃×C₃₃ — C₃²×C₃₃

Lower central

C₁ — C₃²×C₃₃

Upper central

C₁ — C₃²×C₃₃

Generators and relations for C₃²×C₃₃
G = < a,b,c | a³=b³=c³³=1, ab=ba, ac=ca, bc=cb >

Subgroups: 56, all normal (4 characteristic)
C₁, C₃, C₃², C₁₁, C₃³, C₃₃, C₃×C₃₃, C₃²×C₃₃
Quotients: C₁, C₃, C₃², C₁₁, C₃³, C₃₃, C₃×C₃₃, C₃²×C₃₃

Smallest permutation representation of C₃²×C₃₃
►Regular action on 297 points

Generators in S₂₉₇

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

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

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

297 irreducible representations

dim	1	1	1	1
type	+
image	C₁	C₃	C₁₁	C₃₃
kernel	C₃²×C₃₃	C₃×C₃₃	C₃³	C₃²
# reps	1	26	10	260

Matrix representation of C₃²×C₃₃ ►in GL₃(𝔽₆₇) generated by

37	0	0
0	1	0
0	0	1

37	0	0
0	37	0
0	0	29

26	0	0
0	17	0
0	0	22

G:=sub<GL(3,GF(67))| [37,0,0,0,1,0,0,0,1],[37,0,0,0,37,0,0,0,29],[26,0,0,0,17,0,0,0,22] >;

C₃²×C₃₃ in GAP, Magma, Sage, TeX

C_3^2\times C_{33}

% in TeX

G:=Group("C3^2xC33");

// GroupNames label

G:=SmallGroup(297,5);

// by ID

G=gap.SmallGroup(297,5);

# by ID

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

// Polycyclic

G:=Group<a,b,c|a^3=b^3=c^33=1,a*b=b*a,a*c=c*a,b*c=c*b>;

// generators/relations

G = C32×C33 order 297 = 33·11

Abelian group of type [3,3,33]

G = C₃²×C₃₃ order 297 = 3³·11