direct product, abelian, monomial, 2-elementary
Aliases: C23×C26, SmallGroup(208,51)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C23×C26 |
| C1 — C23×C26 |
| C1 — C23×C26 |
Generators and relations for C23×C26
G = < a,b,c,d | a2=b2=c2=d26=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >
Subgroups: 134, all normal (4 characteristic)
C1, C2, C22, C23, C13, C24, C26, C2×C26, C22×C26, C23×C26
Quotients: C1, C2, C22, C23, C13, C24, C26, C2×C26, C22×C26, C23×C26
►Regular action on 208 points
Generators in S208
(1 122)(2 123)(3 124)(4 125)(5 126)(6 127)(7 128)(8 129)(9 130)(10 105)(11 106)(12 107)(13 108)(14 109)(15 110)(16 111)(17 112)(18 113)(19 114)(20 115)(21 116)(22 117)(23 118)(24 119)(25 120)(26 121)(27 137)(28 138)(29 139)(30 140)(31 141)(32 142)(33 143)(34 144)(35 145)(36 146)(37 147)(38 148)(39 149)(40 150)(41 151)(42 152)(43 153)(44 154)(45 155)(46 156)(47 131)(48 132)(49 133)(50 134)(51 135)(52 136)(53 181)(54 182)(55 157)(56 158)(57 159)(58 160)(59 161)(60 162)(61 163)(62 164)(63 165)(64 166)(65 167)(66 168)(67 169)(68 170)(69 171)(70 172)(71 173)(72 174)(73 175)(74 176)(75 177)(76 178)(77 179)(78 180)(79 190)(80 191)(81 192)(82 193)(83 194)(84 195)(85 196)(86 197)(87 198)(88 199)(89 200)(90 201)(91 202)(92 203)(93 204)(94 205)(95 206)(96 207)(97 208)(98 183)(99 184)(100 185)(101 186)(102 187)(103 188)(104 189)
(1 77)(2 78)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 79)(28 80)(29 81)(30 82)(31 83)(32 84)(33 85)(34 86)(35 87)(36 88)(37 89)(38 90)(39 91)(40 92)(41 93)(42 94)(43 95)(44 96)(45 97)(46 98)(47 99)(48 100)(49 101)(50 102)(51 103)(52 104)(105 162)(106 163)(107 164)(108 165)(109 166)(110 167)(111 168)(112 169)(113 170)(114 171)(115 172)(116 173)(117 174)(118 175)(119 176)(120 177)(121 178)(122 179)(123 180)(124 181)(125 182)(126 157)(127 158)(128 159)(129 160)(130 161)(131 184)(132 185)(133 186)(134 187)(135 188)(136 189)(137 190)(138 191)(139 192)(140 193)(141 194)(142 195)(143 196)(144 197)(145 198)(146 199)(147 200)(148 201)(149 202)(150 203)(151 204)(152 205)(153 206)(154 207)(155 208)(156 183)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 27)(14 28)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 39)(26 40)(53 95)(54 96)(55 97)(56 98)(57 99)(58 100)(59 101)(60 102)(61 103)(62 104)(63 79)(64 80)(65 81)(66 82)(67 83)(68 84)(69 85)(70 86)(71 87)(72 88)(73 89)(74 90)(75 91)(76 92)(77 93)(78 94)(105 134)(106 135)(107 136)(108 137)(109 138)(110 139)(111 140)(112 141)(113 142)(114 143)(115 144)(116 145)(117 146)(118 147)(119 148)(120 149)(121 150)(122 151)(123 152)(124 153)(125 154)(126 155)(127 156)(128 131)(129 132)(130 133)(157 208)(158 183)(159 184)(160 185)(161 186)(162 187)(163 188)(164 189)(165 190)(166 191)(167 192)(168 193)(169 194)(170 195)(171 196)(172 197)(173 198)(174 199)(175 200)(176 201)(177 202)(178 203)(179 204)(180 205)(181 206)(182 207)
(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)
G:=sub<Sym(208)| (1,122)(2,123)(3,124)(4,125)(5,126)(6,127)(7,128)(8,129)(9,130)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,115)(21,116)(22,117)(23,118)(24,119)(25,120)(26,121)(27,137)(28,138)(29,139)(30,140)(31,141)(32,142)(33,143)(34,144)(35,145)(36,146)(37,147)(38,148)(39,149)(40,150)(41,151)(42,152)(43,153)(44,154)(45,155)(46,156)(47,131)(48,132)(49,133)(50,134)(51,135)(52,136)(53,181)(54,182)(55,157)(56,158)(57,159)(58,160)(59,161)(60,162)(61,163)(62,164)(63,165)(64,166)(65,167)(66,168)(67,169)(68,170)(69,171)(70,172)(71,173)(72,174)(73,175)(74,176)(75,177)(76,178)(77,179)(78,180)(79,190)(80,191)(81,192)(82,193)(83,194)(84,195)(85,196)(86,197)(87,198)(88,199)(89,200)(90,201)(91,202)(92,203)(93,204)(94,205)(95,206)(96,207)(97,208)(98,183)(99,184)(100,185)(101,186)(102,187)(103,188)(104,189), (1,77)(2,78)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,85)(34,86)(35,87)(36,88)(37,89)(38,90)(39,91)(40,92)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,101)(50,102)(51,103)(52,104)(105,162)(106,163)(107,164)(108,165)(109,166)(110,167)(111,168)(112,169)(113,170)(114,171)(115,172)(116,173)(117,174)(118,175)(119,176)(120,177)(121,178)(122,179)(123,180)(124,181)(125,182)(126,157)(127,158)(128,159)(129,160)(130,161)(131,184)(132,185)(133,186)(134,187)(135,188)(136,189)(137,190)(138,191)(139,192)(140,193)(141,194)(142,195)(143,196)(144,197)(145,198)(146,199)(147,200)(148,201)(149,202)(150,203)(151,204)(152,205)(153,206)(154,207)(155,208)(156,183), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(53,95)(54,96)(55,97)(56,98)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,79)(64,80)(65,81)(66,82)(67,83)(68,84)(69,85)(70,86)(71,87)(72,88)(73,89)(74,90)(75,91)(76,92)(77,93)(78,94)(105,134)(106,135)(107,136)(108,137)(109,138)(110,139)(111,140)(112,141)(113,142)(114,143)(115,144)(116,145)(117,146)(118,147)(119,148)(120,149)(121,150)(122,151)(123,152)(124,153)(125,154)(126,155)(127,156)(128,131)(129,132)(130,133)(157,208)(158,183)(159,184)(160,185)(161,186)(162,187)(163,188)(164,189)(165,190)(166,191)(167,192)(168,193)(169,194)(170,195)(171,196)(172,197)(173,198)(174,199)(175,200)(176,201)(177,202)(178,203)(179,204)(180,205)(181,206)(182,207), (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)>;
G:=Group( (1,122)(2,123)(3,124)(4,125)(5,126)(6,127)(7,128)(8,129)(9,130)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,115)(21,116)(22,117)(23,118)(24,119)(25,120)(26,121)(27,137)(28,138)(29,139)(30,140)(31,141)(32,142)(33,143)(34,144)(35,145)(36,146)(37,147)(38,148)(39,149)(40,150)(41,151)(42,152)(43,153)(44,154)(45,155)(46,156)(47,131)(48,132)(49,133)(50,134)(51,135)(52,136)(53,181)(54,182)(55,157)(56,158)(57,159)(58,160)(59,161)(60,162)(61,163)(62,164)(63,165)(64,166)(65,167)(66,168)(67,169)(68,170)(69,171)(70,172)(71,173)(72,174)(73,175)(74,176)(75,177)(76,178)(77,179)(78,180)(79,190)(80,191)(81,192)(82,193)(83,194)(84,195)(85,196)(86,197)(87,198)(88,199)(89,200)(90,201)(91,202)(92,203)(93,204)(94,205)(95,206)(96,207)(97,208)(98,183)(99,184)(100,185)(101,186)(102,187)(103,188)(104,189), (1,77)(2,78)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,85)(34,86)(35,87)(36,88)(37,89)(38,90)(39,91)(40,92)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,101)(50,102)(51,103)(52,104)(105,162)(106,163)(107,164)(108,165)(109,166)(110,167)(111,168)(112,169)(113,170)(114,171)(115,172)(116,173)(117,174)(118,175)(119,176)(120,177)(121,178)(122,179)(123,180)(124,181)(125,182)(126,157)(127,158)(128,159)(129,160)(130,161)(131,184)(132,185)(133,186)(134,187)(135,188)(136,189)(137,190)(138,191)(139,192)(140,193)(141,194)(142,195)(143,196)(144,197)(145,198)(146,199)(147,200)(148,201)(149,202)(150,203)(151,204)(152,205)(153,206)(154,207)(155,208)(156,183), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(53,95)(54,96)(55,97)(56,98)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,79)(64,80)(65,81)(66,82)(67,83)(68,84)(69,85)(70,86)(71,87)(72,88)(73,89)(74,90)(75,91)(76,92)(77,93)(78,94)(105,134)(106,135)(107,136)(108,137)(109,138)(110,139)(111,140)(112,141)(113,142)(114,143)(115,144)(116,145)(117,146)(118,147)(119,148)(120,149)(121,150)(122,151)(123,152)(124,153)(125,154)(126,155)(127,156)(128,131)(129,132)(130,133)(157,208)(158,183)(159,184)(160,185)(161,186)(162,187)(163,188)(164,189)(165,190)(166,191)(167,192)(168,193)(169,194)(170,195)(171,196)(172,197)(173,198)(174,199)(175,200)(176,201)(177,202)(178,203)(179,204)(180,205)(181,206)(182,207), (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) );
G=PermutationGroup([[(1,122),(2,123),(3,124),(4,125),(5,126),(6,127),(7,128),(8,129),(9,130),(10,105),(11,106),(12,107),(13,108),(14,109),(15,110),(16,111),(17,112),(18,113),(19,114),(20,115),(21,116),(22,117),(23,118),(24,119),(25,120),(26,121),(27,137),(28,138),(29,139),(30,140),(31,141),(32,142),(33,143),(34,144),(35,145),(36,146),(37,147),(38,148),(39,149),(40,150),(41,151),(42,152),(43,153),(44,154),(45,155),(46,156),(47,131),(48,132),(49,133),(50,134),(51,135),(52,136),(53,181),(54,182),(55,157),(56,158),(57,159),(58,160),(59,161),(60,162),(61,163),(62,164),(63,165),(64,166),(65,167),(66,168),(67,169),(68,170),(69,171),(70,172),(71,173),(72,174),(73,175),(74,176),(75,177),(76,178),(77,179),(78,180),(79,190),(80,191),(81,192),(82,193),(83,194),(84,195),(85,196),(86,197),(87,198),(88,199),(89,200),(90,201),(91,202),(92,203),(93,204),(94,205),(95,206),(96,207),(97,208),(98,183),(99,184),(100,185),(101,186),(102,187),(103,188),(104,189)], [(1,77),(2,78),(3,53),(4,54),(5,55),(6,56),(7,57),(8,58),(9,59),(10,60),(11,61),(12,62),(13,63),(14,64),(15,65),(16,66),(17,67),(18,68),(19,69),(20,70),(21,71),(22,72),(23,73),(24,74),(25,75),(26,76),(27,79),(28,80),(29,81),(30,82),(31,83),(32,84),(33,85),(34,86),(35,87),(36,88),(37,89),(38,90),(39,91),(40,92),(41,93),(42,94),(43,95),(44,96),(45,97),(46,98),(47,99),(48,100),(49,101),(50,102),(51,103),(52,104),(105,162),(106,163),(107,164),(108,165),(109,166),(110,167),(111,168),(112,169),(113,170),(114,171),(115,172),(116,173),(117,174),(118,175),(119,176),(120,177),(121,178),(122,179),(123,180),(124,181),(125,182),(126,157),(127,158),(128,159),(129,160),(130,161),(131,184),(132,185),(133,186),(134,187),(135,188),(136,189),(137,190),(138,191),(139,192),(140,193),(141,194),(142,195),(143,196),(144,197),(145,198),(146,199),(147,200),(148,201),(149,202),(150,203),(151,204),(152,205),(153,206),(154,207),(155,208),(156,183)], [(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,27),(14,28),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,39),(26,40),(53,95),(54,96),(55,97),(56,98),(57,99),(58,100),(59,101),(60,102),(61,103),(62,104),(63,79),(64,80),(65,81),(66,82),(67,83),(68,84),(69,85),(70,86),(71,87),(72,88),(73,89),(74,90),(75,91),(76,92),(77,93),(78,94),(105,134),(106,135),(107,136),(108,137),(109,138),(110,139),(111,140),(112,141),(113,142),(114,143),(115,144),(116,145),(117,146),(118,147),(119,148),(120,149),(121,150),(122,151),(123,152),(124,153),(125,154),(126,155),(127,156),(128,131),(129,132),(130,133),(157,208),(158,183),(159,184),(160,185),(161,186),(162,187),(163,188),(164,189),(165,190),(166,191),(167,192),(168,193),(169,194),(170,195),(171,196),(172,197),(173,198),(174,199),(175,200),(176,201),(177,202),(178,203),(179,204),(180,205),(181,206),(182,207)], [(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)]])
C23×C26 is a maximal subgroup of
C24⋊D13
208 conjugacy classes
| class | 1 | 2A | ··· | 2O | 13A | ··· | 13L | 26A | ··· | 26FX |
| order | 1 | 2 | ··· | 2 | 13 | ··· | 13 | 26 | ··· | 26 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
208 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C13 | C26 |
| kernel | C23×C26 | C22×C26 | C24 | C23 |
| # reps | 1 | 15 | 12 | 180 |
Matrix representation of C23×C26 ►in GL4(𝔽53) generated by
| 52 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 52 |
| 52 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 52 | 0 | 0 | 0 |
| 0 | 38 | 0 | 0 |
| 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 47 |
G:=sub<GL(4,GF(53))| [52,0,0,0,0,1,0,0,0,0,52,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,52],[52,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[52,0,0,0,0,38,0,0,0,0,17,0,0,0,0,47] >;
C23×C26 in GAP, Magma, Sage, TeX
C_2^3\times C_{26} % in TeX
G:=Group("C2^3xC26"); // GroupNames label
G:=SmallGroup(208,51);
// by ID
G=gap.SmallGroup(208,51);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^2=d^26=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations