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G = C32×C33order 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

C1 — C32×C33
C1C11C33C3×C33 — C32×C33
C1 — C32×C33
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 [×13], C32 [×13], C11, C33, C33 [×13], C3×C33 [×13], C32×C33
Quotients: C1, C3 [×13], C32 [×13], C11, C33, C33 [×13], C3×C33 [×13], C32×C33

Smallest permutation representation of C32×C33
Regular action on 297 points
Generators in S297
(1 216 113)(2 217 114)(3 218 115)(4 219 116)(5 220 117)(6 221 118)(7 222 119)(8 223 120)(9 224 121)(10 225 122)(11 226 123)(12 227 124)(13 228 125)(14 229 126)(15 230 127)(16 231 128)(17 199 129)(18 200 130)(19 201 131)(20 202 132)(21 203 100)(22 204 101)(23 205 102)(24 206 103)(25 207 104)(26 208 105)(27 209 106)(28 210 107)(29 211 108)(30 212 109)(31 213 110)(32 214 111)(33 215 112)(34 257 160)(35 258 161)(36 259 162)(37 260 163)(38 261 164)(39 262 165)(40 263 133)(41 264 134)(42 232 135)(43 233 136)(44 234 137)(45 235 138)(46 236 139)(47 237 140)(48 238 141)(49 239 142)(50 240 143)(51 241 144)(52 242 145)(53 243 146)(54 244 147)(55 245 148)(56 246 149)(57 247 150)(58 248 151)(59 249 152)(60 250 153)(61 251 154)(62 252 155)(63 253 156)(64 254 157)(65 255 158)(66 256 159)(67 288 166)(68 289 167)(69 290 168)(70 291 169)(71 292 170)(72 293 171)(73 294 172)(74 295 173)(75 296 174)(76 297 175)(77 265 176)(78 266 177)(79 267 178)(80 268 179)(81 269 180)(82 270 181)(83 271 182)(84 272 183)(85 273 184)(86 274 185)(87 275 186)(88 276 187)(89 277 188)(90 278 189)(91 279 190)(92 280 191)(93 281 192)(94 282 193)(95 283 194)(96 284 195)(97 285 196)(98 286 197)(99 287 198)
(1 90 61)(2 91 62)(3 92 63)(4 93 64)(5 94 65)(6 95 66)(7 96 34)(8 97 35)(9 98 36)(10 99 37)(11 67 38)(12 68 39)(13 69 40)(14 70 41)(15 71 42)(16 72 43)(17 73 44)(18 74 45)(19 75 46)(20 76 47)(21 77 48)(22 78 49)(23 79 50)(24 80 51)(25 81 52)(26 82 53)(27 83 54)(28 84 55)(29 85 56)(30 86 57)(31 87 58)(32 88 59)(33 89 60)(100 176 141)(101 177 142)(102 178 143)(103 179 144)(104 180 145)(105 181 146)(106 182 147)(107 183 148)(108 184 149)(109 185 150)(110 186 151)(111 187 152)(112 188 153)(113 189 154)(114 190 155)(115 191 156)(116 192 157)(117 193 158)(118 194 159)(119 195 160)(120 196 161)(121 197 162)(122 198 163)(123 166 164)(124 167 165)(125 168 133)(126 169 134)(127 170 135)(128 171 136)(129 172 137)(130 173 138)(131 174 139)(132 175 140)(199 294 234)(200 295 235)(201 296 236)(202 297 237)(203 265 238)(204 266 239)(205 267 240)(206 268 241)(207 269 242)(208 270 243)(209 271 244)(210 272 245)(211 273 246)(212 274 247)(213 275 248)(214 276 249)(215 277 250)(216 278 251)(217 279 252)(218 280 253)(219 281 254)(220 282 255)(221 283 256)(222 284 257)(223 285 258)(224 286 259)(225 287 260)(226 288 261)(227 289 262)(228 290 263)(229 291 264)(230 292 232)(231 293 233)
(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)| (1,216,113)(2,217,114)(3,218,115)(4,219,116)(5,220,117)(6,221,118)(7,222,119)(8,223,120)(9,224,121)(10,225,122)(11,226,123)(12,227,124)(13,228,125)(14,229,126)(15,230,127)(16,231,128)(17,199,129)(18,200,130)(19,201,131)(20,202,132)(21,203,100)(22,204,101)(23,205,102)(24,206,103)(25,207,104)(26,208,105)(27,209,106)(28,210,107)(29,211,108)(30,212,109)(31,213,110)(32,214,111)(33,215,112)(34,257,160)(35,258,161)(36,259,162)(37,260,163)(38,261,164)(39,262,165)(40,263,133)(41,264,134)(42,232,135)(43,233,136)(44,234,137)(45,235,138)(46,236,139)(47,237,140)(48,238,141)(49,239,142)(50,240,143)(51,241,144)(52,242,145)(53,243,146)(54,244,147)(55,245,148)(56,246,149)(57,247,150)(58,248,151)(59,249,152)(60,250,153)(61,251,154)(62,252,155)(63,253,156)(64,254,157)(65,255,158)(66,256,159)(67,288,166)(68,289,167)(69,290,168)(70,291,169)(71,292,170)(72,293,171)(73,294,172)(74,295,173)(75,296,174)(76,297,175)(77,265,176)(78,266,177)(79,267,178)(80,268,179)(81,269,180)(82,270,181)(83,271,182)(84,272,183)(85,273,184)(86,274,185)(87,275,186)(88,276,187)(89,277,188)(90,278,189)(91,279,190)(92,280,191)(93,281,192)(94,282,193)(95,283,194)(96,284,195)(97,285,196)(98,286,197)(99,287,198), (1,90,61)(2,91,62)(3,92,63)(4,93,64)(5,94,65)(6,95,66)(7,96,34)(8,97,35)(9,98,36)(10,99,37)(11,67,38)(12,68,39)(13,69,40)(14,70,41)(15,71,42)(16,72,43)(17,73,44)(18,74,45)(19,75,46)(20,76,47)(21,77,48)(22,78,49)(23,79,50)(24,80,51)(25,81,52)(26,82,53)(27,83,54)(28,84,55)(29,85,56)(30,86,57)(31,87,58)(32,88,59)(33,89,60)(100,176,141)(101,177,142)(102,178,143)(103,179,144)(104,180,145)(105,181,146)(106,182,147)(107,183,148)(108,184,149)(109,185,150)(110,186,151)(111,187,152)(112,188,153)(113,189,154)(114,190,155)(115,191,156)(116,192,157)(117,193,158)(118,194,159)(119,195,160)(120,196,161)(121,197,162)(122,198,163)(123,166,164)(124,167,165)(125,168,133)(126,169,134)(127,170,135)(128,171,136)(129,172,137)(130,173,138)(131,174,139)(132,175,140)(199,294,234)(200,295,235)(201,296,236)(202,297,237)(203,265,238)(204,266,239)(205,267,240)(206,268,241)(207,269,242)(208,270,243)(209,271,244)(210,272,245)(211,273,246)(212,274,247)(213,275,248)(214,276,249)(215,277,250)(216,278,251)(217,279,252)(218,280,253)(219,281,254)(220,282,255)(221,283,256)(222,284,257)(223,285,258)(224,286,259)(225,287,260)(226,288,261)(227,289,262)(228,290,263)(229,291,264)(230,292,232)(231,293,233), (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( (1,216,113)(2,217,114)(3,218,115)(4,219,116)(5,220,117)(6,221,118)(7,222,119)(8,223,120)(9,224,121)(10,225,122)(11,226,123)(12,227,124)(13,228,125)(14,229,126)(15,230,127)(16,231,128)(17,199,129)(18,200,130)(19,201,131)(20,202,132)(21,203,100)(22,204,101)(23,205,102)(24,206,103)(25,207,104)(26,208,105)(27,209,106)(28,210,107)(29,211,108)(30,212,109)(31,213,110)(32,214,111)(33,215,112)(34,257,160)(35,258,161)(36,259,162)(37,260,163)(38,261,164)(39,262,165)(40,263,133)(41,264,134)(42,232,135)(43,233,136)(44,234,137)(45,235,138)(46,236,139)(47,237,140)(48,238,141)(49,239,142)(50,240,143)(51,241,144)(52,242,145)(53,243,146)(54,244,147)(55,245,148)(56,246,149)(57,247,150)(58,248,151)(59,249,152)(60,250,153)(61,251,154)(62,252,155)(63,253,156)(64,254,157)(65,255,158)(66,256,159)(67,288,166)(68,289,167)(69,290,168)(70,291,169)(71,292,170)(72,293,171)(73,294,172)(74,295,173)(75,296,174)(76,297,175)(77,265,176)(78,266,177)(79,267,178)(80,268,179)(81,269,180)(82,270,181)(83,271,182)(84,272,183)(85,273,184)(86,274,185)(87,275,186)(88,276,187)(89,277,188)(90,278,189)(91,279,190)(92,280,191)(93,281,192)(94,282,193)(95,283,194)(96,284,195)(97,285,196)(98,286,197)(99,287,198), (1,90,61)(2,91,62)(3,92,63)(4,93,64)(5,94,65)(6,95,66)(7,96,34)(8,97,35)(9,98,36)(10,99,37)(11,67,38)(12,68,39)(13,69,40)(14,70,41)(15,71,42)(16,72,43)(17,73,44)(18,74,45)(19,75,46)(20,76,47)(21,77,48)(22,78,49)(23,79,50)(24,80,51)(25,81,52)(26,82,53)(27,83,54)(28,84,55)(29,85,56)(30,86,57)(31,87,58)(32,88,59)(33,89,60)(100,176,141)(101,177,142)(102,178,143)(103,179,144)(104,180,145)(105,181,146)(106,182,147)(107,183,148)(108,184,149)(109,185,150)(110,186,151)(111,187,152)(112,188,153)(113,189,154)(114,190,155)(115,191,156)(116,192,157)(117,193,158)(118,194,159)(119,195,160)(120,196,161)(121,197,162)(122,198,163)(123,166,164)(124,167,165)(125,168,133)(126,169,134)(127,170,135)(128,171,136)(129,172,137)(130,173,138)(131,174,139)(132,175,140)(199,294,234)(200,295,235)(201,296,236)(202,297,237)(203,265,238)(204,266,239)(205,267,240)(206,268,241)(207,269,242)(208,270,243)(209,271,244)(210,272,245)(211,273,246)(212,274,247)(213,275,248)(214,276,249)(215,277,250)(216,278,251)(217,279,252)(218,280,253)(219,281,254)(220,282,255)(221,283,256)(222,284,257)(223,285,258)(224,286,259)(225,287,260)(226,288,261)(227,289,262)(228,290,263)(229,291,264)(230,292,232)(231,293,233), (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([(1,216,113),(2,217,114),(3,218,115),(4,219,116),(5,220,117),(6,221,118),(7,222,119),(8,223,120),(9,224,121),(10,225,122),(11,226,123),(12,227,124),(13,228,125),(14,229,126),(15,230,127),(16,231,128),(17,199,129),(18,200,130),(19,201,131),(20,202,132),(21,203,100),(22,204,101),(23,205,102),(24,206,103),(25,207,104),(26,208,105),(27,209,106),(28,210,107),(29,211,108),(30,212,109),(31,213,110),(32,214,111),(33,215,112),(34,257,160),(35,258,161),(36,259,162),(37,260,163),(38,261,164),(39,262,165),(40,263,133),(41,264,134),(42,232,135),(43,233,136),(44,234,137),(45,235,138),(46,236,139),(47,237,140),(48,238,141),(49,239,142),(50,240,143),(51,241,144),(52,242,145),(53,243,146),(54,244,147),(55,245,148),(56,246,149),(57,247,150),(58,248,151),(59,249,152),(60,250,153),(61,251,154),(62,252,155),(63,253,156),(64,254,157),(65,255,158),(66,256,159),(67,288,166),(68,289,167),(69,290,168),(70,291,169),(71,292,170),(72,293,171),(73,294,172),(74,295,173),(75,296,174),(76,297,175),(77,265,176),(78,266,177),(79,267,178),(80,268,179),(81,269,180),(82,270,181),(83,271,182),(84,272,183),(85,273,184),(86,274,185),(87,275,186),(88,276,187),(89,277,188),(90,278,189),(91,279,190),(92,280,191),(93,281,192),(94,282,193),(95,283,194),(96,284,195),(97,285,196),(98,286,197),(99,287,198)], [(1,90,61),(2,91,62),(3,92,63),(4,93,64),(5,94,65),(6,95,66),(7,96,34),(8,97,35),(9,98,36),(10,99,37),(11,67,38),(12,68,39),(13,69,40),(14,70,41),(15,71,42),(16,72,43),(17,73,44),(18,74,45),(19,75,46),(20,76,47),(21,77,48),(22,78,49),(23,79,50),(24,80,51),(25,81,52),(26,82,53),(27,83,54),(28,84,55),(29,85,56),(30,86,57),(31,87,58),(32,88,59),(33,89,60),(100,176,141),(101,177,142),(102,178,143),(103,179,144),(104,180,145),(105,181,146),(106,182,147),(107,183,148),(108,184,149),(109,185,150),(110,186,151),(111,187,152),(112,188,153),(113,189,154),(114,190,155),(115,191,156),(116,192,157),(117,193,158),(118,194,159),(119,195,160),(120,196,161),(121,197,162),(122,198,163),(123,166,164),(124,167,165),(125,168,133),(126,169,134),(127,170,135),(128,171,136),(129,172,137),(130,173,138),(131,174,139),(132,175,140),(199,294,234),(200,295,235),(201,296,236),(202,297,237),(203,265,238),(204,266,239),(205,267,240),(206,268,241),(207,269,242),(208,270,243),(209,271,244),(210,272,245),(211,273,246),(212,274,247),(213,275,248),(214,276,249),(215,277,250),(216,278,251),(217,279,252),(218,280,253),(219,281,254),(220,282,255),(221,283,256),(222,284,257),(223,285,258),(224,286,259),(225,287,260),(226,288,261),(227,289,262),(228,290,263),(229,291,264),(230,292,232),(231,293,233)], [(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···3Z11A···11J33A···33IZ
order13···311···1133···33
size11···11···11···1

297 irreducible representations

dim1111
type+
imageC1C3C11C33
kernelC32×C33C3×C33C33C32
# reps12610260

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

3700
010
001
,
3700
0370
0029
,
2600
0170
0022
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

׿
×
𝔽