direct product, abelian, monomial, 2-elementary
Aliases: C22×C66, SmallGroup(264,39)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C66 |
| C1 — C22×C66 |
| C1 — C22×C66 |
Generators and relations for C22×C66
G = < a,b,c | a2=b2=c66=1, ab=ba, ac=ca, bc=cb >
Subgroups: 64, all normal (8 characteristic)
C1, C2, C3, C22, C6, C23, C11, C2×C6, C22, C22×C6, C33, C2×C22, C66, C22×C22, C2×C66, C22×C66
Quotients: C1, C2, C3, C22, C6, C23, C11, C2×C6, C22, C22×C6, C33, C2×C22, C66, C22×C22, C2×C66, C22×C66
►Regular action on 264 points
Generators in S264
(1 192)(2 193)(3 194)(4 195)(5 196)(6 197)(7 198)(8 133)(9 134)(10 135)(11 136)(12 137)(13 138)(14 139)(15 140)(16 141)(17 142)(18 143)(19 144)(20 145)(21 146)(22 147)(23 148)(24 149)(25 150)(26 151)(27 152)(28 153)(29 154)(30 155)(31 156)(32 157)(33 158)(34 159)(35 160)(36 161)(37 162)(38 163)(39 164)(40 165)(41 166)(42 167)(43 168)(44 169)(45 170)(46 171)(47 172)(48 173)(49 174)(50 175)(51 176)(52 177)(53 178)(54 179)(55 180)(56 181)(57 182)(58 183)(59 184)(60 185)(61 186)(62 187)(63 188)(64 189)(65 190)(66 191)(67 202)(68 203)(69 204)(70 205)(71 206)(72 207)(73 208)(74 209)(75 210)(76 211)(77 212)(78 213)(79 214)(80 215)(81 216)(82 217)(83 218)(84 219)(85 220)(86 221)(87 222)(88 223)(89 224)(90 225)(91 226)(92 227)(93 228)(94 229)(95 230)(96 231)(97 232)(98 233)(99 234)(100 235)(101 236)(102 237)(103 238)(104 239)(105 240)(106 241)(107 242)(108 243)(109 244)(110 245)(111 246)(112 247)(113 248)(114 249)(115 250)(116 251)(117 252)(118 253)(119 254)(120 255)(121 256)(122 257)(123 258)(124 259)(125 260)(126 261)(127 262)(128 263)(129 264)(130 199)(131 200)(132 201)
(1 118)(2 119)(3 120)(4 121)(5 122)(6 123)(7 124)(8 125)(9 126)(10 127)(11 128)(12 129)(13 130)(14 131)(15 132)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 80)(30 81)(31 82)(32 83)(33 84)(34 85)(35 86)(36 87)(37 88)(38 89)(39 90)(40 91)(41 92)(42 93)(43 94)(44 95)(45 96)(46 97)(47 98)(48 99)(49 100)(50 101)(51 102)(52 103)(53 104)(54 105)(55 106)(56 107)(57 108)(58 109)(59 110)(60 111)(61 112)(62 113)(63 114)(64 115)(65 116)(66 117)(133 260)(134 261)(135 262)(136 263)(137 264)(138 199)(139 200)(140 201)(141 202)(142 203)(143 204)(144 205)(145 206)(146 207)(147 208)(148 209)(149 210)(150 211)(151 212)(152 213)(153 214)(154 215)(155 216)(156 217)(157 218)(158 219)(159 220)(160 221)(161 222)(162 223)(163 224)(164 225)(165 226)(166 227)(167 228)(168 229)(169 230)(170 231)(171 232)(172 233)(173 234)(174 235)(175 236)(176 237)(177 238)(178 239)(179 240)(180 241)(181 242)(182 243)(183 244)(184 245)(185 246)(186 247)(187 248)(188 249)(189 250)(190 251)(191 252)(192 253)(193 254)(194 255)(195 256)(196 257)(197 258)(198 259)
(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)
G:=sub<Sym(264)| (1,192)(2,193)(3,194)(4,195)(5,196)(6,197)(7,198)(8,133)(9,134)(10,135)(11,136)(12,137)(13,138)(14,139)(15,140)(16,141)(17,142)(18,143)(19,144)(20,145)(21,146)(22,147)(23,148)(24,149)(25,150)(26,151)(27,152)(28,153)(29,154)(30,155)(31,156)(32,157)(33,158)(34,159)(35,160)(36,161)(37,162)(38,163)(39,164)(40,165)(41,166)(42,167)(43,168)(44,169)(45,170)(46,171)(47,172)(48,173)(49,174)(50,175)(51,176)(52,177)(53,178)(54,179)(55,180)(56,181)(57,182)(58,183)(59,184)(60,185)(61,186)(62,187)(63,188)(64,189)(65,190)(66,191)(67,202)(68,203)(69,204)(70,205)(71,206)(72,207)(73,208)(74,209)(75,210)(76,211)(77,212)(78,213)(79,214)(80,215)(81,216)(82,217)(83,218)(84,219)(85,220)(86,221)(87,222)(88,223)(89,224)(90,225)(91,226)(92,227)(93,228)(94,229)(95,230)(96,231)(97,232)(98,233)(99,234)(100,235)(101,236)(102,237)(103,238)(104,239)(105,240)(106,241)(107,242)(108,243)(109,244)(110,245)(111,246)(112,247)(113,248)(114,249)(115,250)(116,251)(117,252)(118,253)(119,254)(120,255)(121,256)(122,257)(123,258)(124,259)(125,260)(126,261)(127,262)(128,263)(129,264)(130,199)(131,200)(132,201), (1,118)(2,119)(3,120)(4,121)(5,122)(6,123)(7,124)(8,125)(9,126)(10,127)(11,128)(12,129)(13,130)(14,131)(15,132)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,81)(31,82)(32,83)(33,84)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,97)(47,98)(48,99)(49,100)(50,101)(51,102)(52,103)(53,104)(54,105)(55,106)(56,107)(57,108)(58,109)(59,110)(60,111)(61,112)(62,113)(63,114)(64,115)(65,116)(66,117)(133,260)(134,261)(135,262)(136,263)(137,264)(138,199)(139,200)(140,201)(141,202)(142,203)(143,204)(144,205)(145,206)(146,207)(147,208)(148,209)(149,210)(150,211)(151,212)(152,213)(153,214)(154,215)(155,216)(156,217)(157,218)(158,219)(159,220)(160,221)(161,222)(162,223)(163,224)(164,225)(165,226)(166,227)(167,228)(168,229)(169,230)(170,231)(171,232)(172,233)(173,234)(174,235)(175,236)(176,237)(177,238)(178,239)(179,240)(180,241)(181,242)(182,243)(183,244)(184,245)(185,246)(186,247)(187,248)(188,249)(189,250)(190,251)(191,252)(192,253)(193,254)(194,255)(195,256)(196,257)(197,258)(198,259), (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)>;
G:=Group( (1,192)(2,193)(3,194)(4,195)(5,196)(6,197)(7,198)(8,133)(9,134)(10,135)(11,136)(12,137)(13,138)(14,139)(15,140)(16,141)(17,142)(18,143)(19,144)(20,145)(21,146)(22,147)(23,148)(24,149)(25,150)(26,151)(27,152)(28,153)(29,154)(30,155)(31,156)(32,157)(33,158)(34,159)(35,160)(36,161)(37,162)(38,163)(39,164)(40,165)(41,166)(42,167)(43,168)(44,169)(45,170)(46,171)(47,172)(48,173)(49,174)(50,175)(51,176)(52,177)(53,178)(54,179)(55,180)(56,181)(57,182)(58,183)(59,184)(60,185)(61,186)(62,187)(63,188)(64,189)(65,190)(66,191)(67,202)(68,203)(69,204)(70,205)(71,206)(72,207)(73,208)(74,209)(75,210)(76,211)(77,212)(78,213)(79,214)(80,215)(81,216)(82,217)(83,218)(84,219)(85,220)(86,221)(87,222)(88,223)(89,224)(90,225)(91,226)(92,227)(93,228)(94,229)(95,230)(96,231)(97,232)(98,233)(99,234)(100,235)(101,236)(102,237)(103,238)(104,239)(105,240)(106,241)(107,242)(108,243)(109,244)(110,245)(111,246)(112,247)(113,248)(114,249)(115,250)(116,251)(117,252)(118,253)(119,254)(120,255)(121,256)(122,257)(123,258)(124,259)(125,260)(126,261)(127,262)(128,263)(129,264)(130,199)(131,200)(132,201), (1,118)(2,119)(3,120)(4,121)(5,122)(6,123)(7,124)(8,125)(9,126)(10,127)(11,128)(12,129)(13,130)(14,131)(15,132)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,81)(31,82)(32,83)(33,84)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,97)(47,98)(48,99)(49,100)(50,101)(51,102)(52,103)(53,104)(54,105)(55,106)(56,107)(57,108)(58,109)(59,110)(60,111)(61,112)(62,113)(63,114)(64,115)(65,116)(66,117)(133,260)(134,261)(135,262)(136,263)(137,264)(138,199)(139,200)(140,201)(141,202)(142,203)(143,204)(144,205)(145,206)(146,207)(147,208)(148,209)(149,210)(150,211)(151,212)(152,213)(153,214)(154,215)(155,216)(156,217)(157,218)(158,219)(159,220)(160,221)(161,222)(162,223)(163,224)(164,225)(165,226)(166,227)(167,228)(168,229)(169,230)(170,231)(171,232)(172,233)(173,234)(174,235)(175,236)(176,237)(177,238)(178,239)(179,240)(180,241)(181,242)(182,243)(183,244)(184,245)(185,246)(186,247)(187,248)(188,249)(189,250)(190,251)(191,252)(192,253)(193,254)(194,255)(195,256)(196,257)(197,258)(198,259), (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) );
G=PermutationGroup([[(1,192),(2,193),(3,194),(4,195),(5,196),(6,197),(7,198),(8,133),(9,134),(10,135),(11,136),(12,137),(13,138),(14,139),(15,140),(16,141),(17,142),(18,143),(19,144),(20,145),(21,146),(22,147),(23,148),(24,149),(25,150),(26,151),(27,152),(28,153),(29,154),(30,155),(31,156),(32,157),(33,158),(34,159),(35,160),(36,161),(37,162),(38,163),(39,164),(40,165),(41,166),(42,167),(43,168),(44,169),(45,170),(46,171),(47,172),(48,173),(49,174),(50,175),(51,176),(52,177),(53,178),(54,179),(55,180),(56,181),(57,182),(58,183),(59,184),(60,185),(61,186),(62,187),(63,188),(64,189),(65,190),(66,191),(67,202),(68,203),(69,204),(70,205),(71,206),(72,207),(73,208),(74,209),(75,210),(76,211),(77,212),(78,213),(79,214),(80,215),(81,216),(82,217),(83,218),(84,219),(85,220),(86,221),(87,222),(88,223),(89,224),(90,225),(91,226),(92,227),(93,228),(94,229),(95,230),(96,231),(97,232),(98,233),(99,234),(100,235),(101,236),(102,237),(103,238),(104,239),(105,240),(106,241),(107,242),(108,243),(109,244),(110,245),(111,246),(112,247),(113,248),(114,249),(115,250),(116,251),(117,252),(118,253),(119,254),(120,255),(121,256),(122,257),(123,258),(124,259),(125,260),(126,261),(127,262),(128,263),(129,264),(130,199),(131,200),(132,201)], [(1,118),(2,119),(3,120),(4,121),(5,122),(6,123),(7,124),(8,125),(9,126),(10,127),(11,128),(12,129),(13,130),(14,131),(15,132),(16,67),(17,68),(18,69),(19,70),(20,71),(21,72),(22,73),(23,74),(24,75),(25,76),(26,77),(27,78),(28,79),(29,80),(30,81),(31,82),(32,83),(33,84),(34,85),(35,86),(36,87),(37,88),(38,89),(39,90),(40,91),(41,92),(42,93),(43,94),(44,95),(45,96),(46,97),(47,98),(48,99),(49,100),(50,101),(51,102),(52,103),(53,104),(54,105),(55,106),(56,107),(57,108),(58,109),(59,110),(60,111),(61,112),(62,113),(63,114),(64,115),(65,116),(66,117),(133,260),(134,261),(135,262),(136,263),(137,264),(138,199),(139,200),(140,201),(141,202),(142,203),(143,204),(144,205),(145,206),(146,207),(147,208),(148,209),(149,210),(150,211),(151,212),(152,213),(153,214),(154,215),(155,216),(156,217),(157,218),(158,219),(159,220),(160,221),(161,222),(162,223),(163,224),(164,225),(165,226),(166,227),(167,228),(168,229),(169,230),(170,231),(171,232),(172,233),(173,234),(174,235),(175,236),(176,237),(177,238),(178,239),(179,240),(180,241),(181,242),(182,243),(183,244),(184,245),(185,246),(186,247),(187,248),(188,249),(189,250),(190,251),(191,252),(192,253),(193,254),(194,255),(195,256),(196,257),(197,258),(198,259)], [(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)]])
264 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 6A | ··· | 6N | 11A | ··· | 11J | 22A | ··· | 22BR | 33A | ··· | 33T | 66A | ··· | 66EJ |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 6 | ··· | 6 | 11 | ··· | 11 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
264 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C11 | C22 | C33 | C66 |
| kernel | C22×C66 | C2×C66 | C22×C22 | C2×C22 | C22×C6 | C2×C6 | C23 | C22 |
| # reps | 1 | 7 | 2 | 14 | 10 | 70 | 20 | 140 |
Matrix representation of C22×C66 ►in GL3(𝔽67) generated by
| 66 | 0 | 0 |
| 0 | 66 | 0 |
| 0 | 0 | 66 |
| 66 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 66 |
| 40 | 0 | 0 |
| 0 | 38 | 0 |
| 0 | 0 | 51 |
G:=sub<GL(3,GF(67))| [66,0,0,0,66,0,0,0,66],[66,0,0,0,1,0,0,0,66],[40,0,0,0,38,0,0,0,51] >;
C22×C66 in GAP, Magma, Sage, TeX
C_2^2\times C_{66} % in TeX
G:=Group("C2^2xC66"); // GroupNames label
G:=SmallGroup(264,39);
// by ID
G=gap.SmallGroup(264,39);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-11]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^66=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations