direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×Dic33, C66⋊1C4, C6⋊Dic11, C22⋊Dic3, C2.2D66, C22.D33, C22.11D6, C6.11D22, C66.11C22, C33⋊7(C2×C4), (C2×C22).S3, (C2×C6).D11, (C2×C66).1C2, C3⋊2(C2×Dic11), C11⋊2(C2×Dic3), SmallGroup(264,26)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — C2×Dic33 |
Generators and relations for C2×Dic33
G = < a,b,c | a2=b66=1, c2=b33, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×Dic33
►Regular action on 264 points
Generators in S264
(1 93)(2 94)(3 95)(4 96)(5 97)(6 98)(7 99)(8 100)(9 101)(10 102)(11 103)(12 104)(13 105)(14 106)(15 107)(16 108)(17 109)(18 110)(19 111)(20 112)(21 113)(22 114)(23 115)(24 116)(25 117)(26 118)(27 119)(28 120)(29 121)(30 122)(31 123)(32 124)(33 125)(34 126)(35 127)(36 128)(37 129)(38 130)(39 131)(40 132)(41 67)(42 68)(43 69)(44 70)(45 71)(46 72)(47 73)(48 74)(49 75)(50 76)(51 77)(52 78)(53 79)(54 80)(55 81)(56 82)(57 83)(58 84)(59 85)(60 86)(61 87)(62 88)(63 89)(64 90)(65 91)(66 92)(133 210)(134 211)(135 212)(136 213)(137 214)(138 215)(139 216)(140 217)(141 218)(142 219)(143 220)(144 221)(145 222)(146 223)(147 224)(148 225)(149 226)(150 227)(151 228)(152 229)(153 230)(154 231)(155 232)(156 233)(157 234)(158 235)(159 236)(160 237)(161 238)(162 239)(163 240)(164 241)(165 242)(166 243)(167 244)(168 245)(169 246)(170 247)(171 248)(172 249)(173 250)(174 251)(175 252)(176 253)(177 254)(178 255)(179 256)(180 257)(181 258)(182 259)(183 260)(184 261)(185 262)(186 263)(187 264)(188 199)(189 200)(190 201)(191 202)(192 203)(193 204)(194 205)(195 206)(196 207)(197 208)(198 209)
(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)
(1 232 34 199)(2 231 35 264)(3 230 36 263)(4 229 37 262)(5 228 38 261)(6 227 39 260)(7 226 40 259)(8 225 41 258)(9 224 42 257)(10 223 43 256)(11 222 44 255)(12 221 45 254)(13 220 46 253)(14 219 47 252)(15 218 48 251)(16 217 49 250)(17 216 50 249)(18 215 51 248)(19 214 52 247)(20 213 53 246)(21 212 54 245)(22 211 55 244)(23 210 56 243)(24 209 57 242)(25 208 58 241)(26 207 59 240)(27 206 60 239)(28 205 61 238)(29 204 62 237)(30 203 63 236)(31 202 64 235)(32 201 65 234)(33 200 66 233)(67 181 100 148)(68 180 101 147)(69 179 102 146)(70 178 103 145)(71 177 104 144)(72 176 105 143)(73 175 106 142)(74 174 107 141)(75 173 108 140)(76 172 109 139)(77 171 110 138)(78 170 111 137)(79 169 112 136)(80 168 113 135)(81 167 114 134)(82 166 115 133)(83 165 116 198)(84 164 117 197)(85 163 118 196)(86 162 119 195)(87 161 120 194)(88 160 121 193)(89 159 122 192)(90 158 123 191)(91 157 124 190)(92 156 125 189)(93 155 126 188)(94 154 127 187)(95 153 128 186)(96 152 129 185)(97 151 130 184)(98 150 131 183)(99 149 132 182)
G:=sub<Sym(264)| (1,93)(2,94)(3,95)(4,96)(5,97)(6,98)(7,99)(8,100)(9,101)(10,102)(11,103)(12,104)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,113)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,120)(29,121)(30,122)(31,123)(32,124)(33,125)(34,126)(35,127)(36,128)(37,129)(38,130)(39,131)(40,132)(41,67)(42,68)(43,69)(44,70)(45,71)(46,72)(47,73)(48,74)(49,75)(50,76)(51,77)(52,78)(53,79)(54,80)(55,81)(56,82)(57,83)(58,84)(59,85)(60,86)(61,87)(62,88)(63,89)(64,90)(65,91)(66,92)(133,210)(134,211)(135,212)(136,213)(137,214)(138,215)(139,216)(140,217)(141,218)(142,219)(143,220)(144,221)(145,222)(146,223)(147,224)(148,225)(149,226)(150,227)(151,228)(152,229)(153,230)(154,231)(155,232)(156,233)(157,234)(158,235)(159,236)(160,237)(161,238)(162,239)(163,240)(164,241)(165,242)(166,243)(167,244)(168,245)(169,246)(170,247)(171,248)(172,249)(173,250)(174,251)(175,252)(176,253)(177,254)(178,255)(179,256)(180,257)(181,258)(182,259)(183,260)(184,261)(185,262)(186,263)(187,264)(188,199)(189,200)(190,201)(191,202)(192,203)(193,204)(194,205)(195,206)(196,207)(197,208)(198,209), (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), (1,232,34,199)(2,231,35,264)(3,230,36,263)(4,229,37,262)(5,228,38,261)(6,227,39,260)(7,226,40,259)(8,225,41,258)(9,224,42,257)(10,223,43,256)(11,222,44,255)(12,221,45,254)(13,220,46,253)(14,219,47,252)(15,218,48,251)(16,217,49,250)(17,216,50,249)(18,215,51,248)(19,214,52,247)(20,213,53,246)(21,212,54,245)(22,211,55,244)(23,210,56,243)(24,209,57,242)(25,208,58,241)(26,207,59,240)(27,206,60,239)(28,205,61,238)(29,204,62,237)(30,203,63,236)(31,202,64,235)(32,201,65,234)(33,200,66,233)(67,181,100,148)(68,180,101,147)(69,179,102,146)(70,178,103,145)(71,177,104,144)(72,176,105,143)(73,175,106,142)(74,174,107,141)(75,173,108,140)(76,172,109,139)(77,171,110,138)(78,170,111,137)(79,169,112,136)(80,168,113,135)(81,167,114,134)(82,166,115,133)(83,165,116,198)(84,164,117,197)(85,163,118,196)(86,162,119,195)(87,161,120,194)(88,160,121,193)(89,159,122,192)(90,158,123,191)(91,157,124,190)(92,156,125,189)(93,155,126,188)(94,154,127,187)(95,153,128,186)(96,152,129,185)(97,151,130,184)(98,150,131,183)(99,149,132,182)>;
G:=Group( (1,93)(2,94)(3,95)(4,96)(5,97)(6,98)(7,99)(8,100)(9,101)(10,102)(11,103)(12,104)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,113)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,120)(29,121)(30,122)(31,123)(32,124)(33,125)(34,126)(35,127)(36,128)(37,129)(38,130)(39,131)(40,132)(41,67)(42,68)(43,69)(44,70)(45,71)(46,72)(47,73)(48,74)(49,75)(50,76)(51,77)(52,78)(53,79)(54,80)(55,81)(56,82)(57,83)(58,84)(59,85)(60,86)(61,87)(62,88)(63,89)(64,90)(65,91)(66,92)(133,210)(134,211)(135,212)(136,213)(137,214)(138,215)(139,216)(140,217)(141,218)(142,219)(143,220)(144,221)(145,222)(146,223)(147,224)(148,225)(149,226)(150,227)(151,228)(152,229)(153,230)(154,231)(155,232)(156,233)(157,234)(158,235)(159,236)(160,237)(161,238)(162,239)(163,240)(164,241)(165,242)(166,243)(167,244)(168,245)(169,246)(170,247)(171,248)(172,249)(173,250)(174,251)(175,252)(176,253)(177,254)(178,255)(179,256)(180,257)(181,258)(182,259)(183,260)(184,261)(185,262)(186,263)(187,264)(188,199)(189,200)(190,201)(191,202)(192,203)(193,204)(194,205)(195,206)(196,207)(197,208)(198,209), (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), (1,232,34,199)(2,231,35,264)(3,230,36,263)(4,229,37,262)(5,228,38,261)(6,227,39,260)(7,226,40,259)(8,225,41,258)(9,224,42,257)(10,223,43,256)(11,222,44,255)(12,221,45,254)(13,220,46,253)(14,219,47,252)(15,218,48,251)(16,217,49,250)(17,216,50,249)(18,215,51,248)(19,214,52,247)(20,213,53,246)(21,212,54,245)(22,211,55,244)(23,210,56,243)(24,209,57,242)(25,208,58,241)(26,207,59,240)(27,206,60,239)(28,205,61,238)(29,204,62,237)(30,203,63,236)(31,202,64,235)(32,201,65,234)(33,200,66,233)(67,181,100,148)(68,180,101,147)(69,179,102,146)(70,178,103,145)(71,177,104,144)(72,176,105,143)(73,175,106,142)(74,174,107,141)(75,173,108,140)(76,172,109,139)(77,171,110,138)(78,170,111,137)(79,169,112,136)(80,168,113,135)(81,167,114,134)(82,166,115,133)(83,165,116,198)(84,164,117,197)(85,163,118,196)(86,162,119,195)(87,161,120,194)(88,160,121,193)(89,159,122,192)(90,158,123,191)(91,157,124,190)(92,156,125,189)(93,155,126,188)(94,154,127,187)(95,153,128,186)(96,152,129,185)(97,151,130,184)(98,150,131,183)(99,149,132,182) );
G=PermutationGroup([[(1,93),(2,94),(3,95),(4,96),(5,97),(6,98),(7,99),(8,100),(9,101),(10,102),(11,103),(12,104),(13,105),(14,106),(15,107),(16,108),(17,109),(18,110),(19,111),(20,112),(21,113),(22,114),(23,115),(24,116),(25,117),(26,118),(27,119),(28,120),(29,121),(30,122),(31,123),(32,124),(33,125),(34,126),(35,127),(36,128),(37,129),(38,130),(39,131),(40,132),(41,67),(42,68),(43,69),(44,70),(45,71),(46,72),(47,73),(48,74),(49,75),(50,76),(51,77),(52,78),(53,79),(54,80),(55,81),(56,82),(57,83),(58,84),(59,85),(60,86),(61,87),(62,88),(63,89),(64,90),(65,91),(66,92),(133,210),(134,211),(135,212),(136,213),(137,214),(138,215),(139,216),(140,217),(141,218),(142,219),(143,220),(144,221),(145,222),(146,223),(147,224),(148,225),(149,226),(150,227),(151,228),(152,229),(153,230),(154,231),(155,232),(156,233),(157,234),(158,235),(159,236),(160,237),(161,238),(162,239),(163,240),(164,241),(165,242),(166,243),(167,244),(168,245),(169,246),(170,247),(171,248),(172,249),(173,250),(174,251),(175,252),(176,253),(177,254),(178,255),(179,256),(180,257),(181,258),(182,259),(183,260),(184,261),(185,262),(186,263),(187,264),(188,199),(189,200),(190,201),(191,202),(192,203),(193,204),(194,205),(195,206),(196,207),(197,208),(198,209)], [(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)], [(1,232,34,199),(2,231,35,264),(3,230,36,263),(4,229,37,262),(5,228,38,261),(6,227,39,260),(7,226,40,259),(8,225,41,258),(9,224,42,257),(10,223,43,256),(11,222,44,255),(12,221,45,254),(13,220,46,253),(14,219,47,252),(15,218,48,251),(16,217,49,250),(17,216,50,249),(18,215,51,248),(19,214,52,247),(20,213,53,246),(21,212,54,245),(22,211,55,244),(23,210,56,243),(24,209,57,242),(25,208,58,241),(26,207,59,240),(27,206,60,239),(28,205,61,238),(29,204,62,237),(30,203,63,236),(31,202,64,235),(32,201,65,234),(33,200,66,233),(67,181,100,148),(68,180,101,147),(69,179,102,146),(70,178,103,145),(71,177,104,144),(72,176,105,143),(73,175,106,142),(74,174,107,141),(75,173,108,140),(76,172,109,139),(77,171,110,138),(78,170,111,137),(79,169,112,136),(80,168,113,135),(81,167,114,134),(82,166,115,133),(83,165,116,198),(84,164,117,197),(85,163,118,196),(86,162,119,195),(87,161,120,194),(88,160,121,193),(89,159,122,192),(90,158,123,191),(91,157,124,190),(92,156,125,189),(93,155,126,188),(94,154,127,187),(95,153,128,186),(96,152,129,185),(97,151,130,184),(98,150,131,183),(99,149,132,182)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 11A | ··· | 11E | 22A | ··· | 22O | 33A | ··· | 33J | 66A | ··· | 66AD |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 11 | ··· | 11 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 |
| size | 1 | 1 | 1 | 1 | 2 | 33 | 33 | 33 | 33 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | + | + | - | + | |
| image | C1 | C2 | C2 | C4 | S3 | Dic3 | D6 | D11 | Dic11 | D22 | D33 | Dic33 | D66 |
| kernel | C2×Dic33 | Dic33 | C2×C66 | C66 | C2×C22 | C22 | C22 | C2×C6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 2 | 1 | 5 | 10 | 5 | 10 | 20 | 10 |
Matrix representation of C2×Dic33 ►in GL3(𝔽397) generated by
| 396 | 0 | 0 |
| 0 | 396 | 0 |
| 0 | 0 | 396 |
| 1 | 0 | 0 |
| 0 | 33 | 321 |
| 0 | 76 | 222 |
| 396 | 0 | 0 |
| 0 | 331 | 27 |
| 0 | 118 | 66 |
G:=sub<GL(3,GF(397))| [396,0,0,0,396,0,0,0,396],[1,0,0,0,33,76,0,321,222],[396,0,0,0,331,118,0,27,66] >;
C2×Dic33 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{33} % in TeX
G:=Group("C2xDic33"); // GroupNames label
G:=SmallGroup(264,26);
// by ID
G=gap.SmallGroup(264,26);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-11,20,323,6004]);
// Polycyclic
G:=Group<a,b,c|a^2=b^66=1,c^2=b^33,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export