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G = C23×C26order 208 = 24·13

Abelian group of type [2,2,2,26]

direct product, abelian, monomial, 2-elementary

Aliases: C23×C26, SmallGroup(208,51)

Series: Derived Chief Lower central Upper central

C1 — C23×C26
C1C13C26C2×C26C22×C26 — 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 [×15], C22 [×35], C23 [×15], C13, C24, C26 [×15], C2×C26 [×35], C22×C26 [×15], C23×C26
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C13, C24, C26 [×15], C2×C26 [×35], C22×C26 [×15], C23×C26

Smallest permutation representation of C23×C26
Regular action on 208 points
Generators in S208
(1 130)(2 105)(3 106)(4 107)(5 108)(6 109)(7 110)(8 111)(9 112)(10 113)(11 114)(12 115)(13 116)(14 117)(15 118)(16 119)(17 120)(18 121)(19 122)(20 123)(21 124)(22 125)(23 126)(24 127)(25 128)(26 129)(27 140)(28 141)(29 142)(30 143)(31 144)(32 145)(33 146)(34 147)(35 148)(36 149)(37 150)(38 151)(39 152)(40 153)(41 154)(42 155)(43 156)(44 131)(45 132)(46 133)(47 134)(48 135)(49 136)(50 137)(51 138)(52 139)(53 182)(54 157)(55 158)(56 159)(57 160)(58 161)(59 162)(60 163)(61 164)(62 165)(63 166)(64 167)(65 168)(66 169)(67 170)(68 171)(69 172)(70 173)(71 174)(72 175)(73 176)(74 177)(75 178)(76 179)(77 180)(78 181)(79 200)(80 201)(81 202)(82 203)(83 204)(84 205)(85 206)(86 207)(87 208)(88 183)(89 184)(90 185)(91 186)(92 187)(93 188)(94 189)(95 190)(96 191)(97 192)(98 193)(99 194)(100 195)(101 196)(102 197)(103 198)(104 199)
(1 74)(2 75)(3 76)(4 77)(5 78)(6 53)(7 54)(8 55)(9 56)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(17 64)(18 65)(19 66)(20 67)(21 68)(22 69)(23 70)(24 71)(25 72)(26 73)(27 88)(28 89)(29 90)(30 91)(31 92)(32 93)(33 94)(34 95)(35 96)(36 97)(37 98)(38 99)(39 100)(40 101)(41 102)(42 103)(43 104)(44 79)(45 80)(46 81)(47 82)(48 83)(49 84)(50 85)(51 86)(52 87)(105 178)(106 179)(107 180)(108 181)(109 182)(110 157)(111 158)(112 159)(113 160)(114 161)(115 162)(116 163)(117 164)(118 165)(119 166)(120 167)(121 168)(122 169)(123 170)(124 171)(125 172)(126 173)(127 174)(128 175)(129 176)(130 177)(131 200)(132 201)(133 202)(134 203)(135 204)(136 205)(137 206)(138 207)(139 208)(140 183)(141 184)(142 185)(143 186)(144 187)(145 188)(146 189)(147 190)(148 191)(149 192)(150 193)(151 194)(152 195)(153 196)(154 197)(155 198)(156 199)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 103)(74 104)(75 79)(76 80)(77 81)(78 82)(105 131)(106 132)(107 133)(108 134)(109 135)(110 136)(111 137)(112 138)(113 139)(114 140)(115 141)(116 142)(117 143)(118 144)(119 145)(120 146)(121 147)(122 148)(123 149)(124 150)(125 151)(126 152)(127 153)(128 154)(129 155)(130 156)(157 205)(158 206)(159 207)(160 208)(161 183)(162 184)(163 185)(164 186)(165 187)(166 188)(167 189)(168 190)(169 191)(170 192)(171 193)(172 194)(173 195)(174 196)(175 197)(176 198)(177 199)(178 200)(179 201)(180 202)(181 203)(182 204)
(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)| (1,130)(2,105)(3,106)(4,107)(5,108)(6,109)(7,110)(8,111)(9,112)(10,113)(11,114)(12,115)(13,116)(14,117)(15,118)(16,119)(17,120)(18,121)(19,122)(20,123)(21,124)(22,125)(23,126)(24,127)(25,128)(26,129)(27,140)(28,141)(29,142)(30,143)(31,144)(32,145)(33,146)(34,147)(35,148)(36,149)(37,150)(38,151)(39,152)(40,153)(41,154)(42,155)(43,156)(44,131)(45,132)(46,133)(47,134)(48,135)(49,136)(50,137)(51,138)(52,139)(53,182)(54,157)(55,158)(56,159)(57,160)(58,161)(59,162)(60,163)(61,164)(62,165)(63,166)(64,167)(65,168)(66,169)(67,170)(68,171)(69,172)(70,173)(71,174)(72,175)(73,176)(74,177)(75,178)(76,179)(77,180)(78,181)(79,200)(80,201)(81,202)(82,203)(83,204)(84,205)(85,206)(86,207)(87,208)(88,183)(89,184)(90,185)(91,186)(92,187)(93,188)(94,189)(95,190)(96,191)(97,192)(98,193)(99,194)(100,195)(101,196)(102,197)(103,198)(104,199), (1,74)(2,75)(3,76)(4,77)(5,78)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,68)(22,69)(23,70)(24,71)(25,72)(26,73)(27,88)(28,89)(29,90)(30,91)(31,92)(32,93)(33,94)(34,95)(35,96)(36,97)(37,98)(38,99)(39,100)(40,101)(41,102)(42,103)(43,104)(44,79)(45,80)(46,81)(47,82)(48,83)(49,84)(50,85)(51,86)(52,87)(105,178)(106,179)(107,180)(108,181)(109,182)(110,157)(111,158)(112,159)(113,160)(114,161)(115,162)(116,163)(117,164)(118,165)(119,166)(120,167)(121,168)(122,169)(123,170)(124,171)(125,172)(126,173)(127,174)(128,175)(129,176)(130,177)(131,200)(132,201)(133,202)(134,203)(135,204)(136,205)(137,206)(138,207)(139,208)(140,183)(141,184)(142,185)(143,186)(144,187)(145,188)(146,189)(147,190)(148,191)(149,192)(150,193)(151,194)(152,195)(153,196)(154,197)(155,198)(156,199), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,103)(74,104)(75,79)(76,80)(77,81)(78,82)(105,131)(106,132)(107,133)(108,134)(109,135)(110,136)(111,137)(112,138)(113,139)(114,140)(115,141)(116,142)(117,143)(118,144)(119,145)(120,146)(121,147)(122,148)(123,149)(124,150)(125,151)(126,152)(127,153)(128,154)(129,155)(130,156)(157,205)(158,206)(159,207)(160,208)(161,183)(162,184)(163,185)(164,186)(165,187)(166,188)(167,189)(168,190)(169,191)(170,192)(171,193)(172,194)(173,195)(174,196)(175,197)(176,198)(177,199)(178,200)(179,201)(180,202)(181,203)(182,204), (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( (1,130)(2,105)(3,106)(4,107)(5,108)(6,109)(7,110)(8,111)(9,112)(10,113)(11,114)(12,115)(13,116)(14,117)(15,118)(16,119)(17,120)(18,121)(19,122)(20,123)(21,124)(22,125)(23,126)(24,127)(25,128)(26,129)(27,140)(28,141)(29,142)(30,143)(31,144)(32,145)(33,146)(34,147)(35,148)(36,149)(37,150)(38,151)(39,152)(40,153)(41,154)(42,155)(43,156)(44,131)(45,132)(46,133)(47,134)(48,135)(49,136)(50,137)(51,138)(52,139)(53,182)(54,157)(55,158)(56,159)(57,160)(58,161)(59,162)(60,163)(61,164)(62,165)(63,166)(64,167)(65,168)(66,169)(67,170)(68,171)(69,172)(70,173)(71,174)(72,175)(73,176)(74,177)(75,178)(76,179)(77,180)(78,181)(79,200)(80,201)(81,202)(82,203)(83,204)(84,205)(85,206)(86,207)(87,208)(88,183)(89,184)(90,185)(91,186)(92,187)(93,188)(94,189)(95,190)(96,191)(97,192)(98,193)(99,194)(100,195)(101,196)(102,197)(103,198)(104,199), (1,74)(2,75)(3,76)(4,77)(5,78)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,68)(22,69)(23,70)(24,71)(25,72)(26,73)(27,88)(28,89)(29,90)(30,91)(31,92)(32,93)(33,94)(34,95)(35,96)(36,97)(37,98)(38,99)(39,100)(40,101)(41,102)(42,103)(43,104)(44,79)(45,80)(46,81)(47,82)(48,83)(49,84)(50,85)(51,86)(52,87)(105,178)(106,179)(107,180)(108,181)(109,182)(110,157)(111,158)(112,159)(113,160)(114,161)(115,162)(116,163)(117,164)(118,165)(119,166)(120,167)(121,168)(122,169)(123,170)(124,171)(125,172)(126,173)(127,174)(128,175)(129,176)(130,177)(131,200)(132,201)(133,202)(134,203)(135,204)(136,205)(137,206)(138,207)(139,208)(140,183)(141,184)(142,185)(143,186)(144,187)(145,188)(146,189)(147,190)(148,191)(149,192)(150,193)(151,194)(152,195)(153,196)(154,197)(155,198)(156,199), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,103)(74,104)(75,79)(76,80)(77,81)(78,82)(105,131)(106,132)(107,133)(108,134)(109,135)(110,136)(111,137)(112,138)(113,139)(114,140)(115,141)(116,142)(117,143)(118,144)(119,145)(120,146)(121,147)(122,148)(123,149)(124,150)(125,151)(126,152)(127,153)(128,154)(129,155)(130,156)(157,205)(158,206)(159,207)(160,208)(161,183)(162,184)(163,185)(164,186)(165,187)(166,188)(167,189)(168,190)(169,191)(170,192)(171,193)(172,194)(173,195)(174,196)(175,197)(176,198)(177,199)(178,200)(179,201)(180,202)(181,203)(182,204), (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([(1,130),(2,105),(3,106),(4,107),(5,108),(6,109),(7,110),(8,111),(9,112),(10,113),(11,114),(12,115),(13,116),(14,117),(15,118),(16,119),(17,120),(18,121),(19,122),(20,123),(21,124),(22,125),(23,126),(24,127),(25,128),(26,129),(27,140),(28,141),(29,142),(30,143),(31,144),(32,145),(33,146),(34,147),(35,148),(36,149),(37,150),(38,151),(39,152),(40,153),(41,154),(42,155),(43,156),(44,131),(45,132),(46,133),(47,134),(48,135),(49,136),(50,137),(51,138),(52,139),(53,182),(54,157),(55,158),(56,159),(57,160),(58,161),(59,162),(60,163),(61,164),(62,165),(63,166),(64,167),(65,168),(66,169),(67,170),(68,171),(69,172),(70,173),(71,174),(72,175),(73,176),(74,177),(75,178),(76,179),(77,180),(78,181),(79,200),(80,201),(81,202),(82,203),(83,204),(84,205),(85,206),(86,207),(87,208),(88,183),(89,184),(90,185),(91,186),(92,187),(93,188),(94,189),(95,190),(96,191),(97,192),(98,193),(99,194),(100,195),(101,196),(102,197),(103,198),(104,199)], [(1,74),(2,75),(3,76),(4,77),(5,78),(6,53),(7,54),(8,55),(9,56),(10,57),(11,58),(12,59),(13,60),(14,61),(15,62),(16,63),(17,64),(18,65),(19,66),(20,67),(21,68),(22,69),(23,70),(24,71),(25,72),(26,73),(27,88),(28,89),(29,90),(30,91),(31,92),(32,93),(33,94),(34,95),(35,96),(36,97),(37,98),(38,99),(39,100),(40,101),(41,102),(42,103),(43,104),(44,79),(45,80),(46,81),(47,82),(48,83),(49,84),(50,85),(51,86),(52,87),(105,178),(106,179),(107,180),(108,181),(109,182),(110,157),(111,158),(112,159),(113,160),(114,161),(115,162),(116,163),(117,164),(118,165),(119,166),(120,167),(121,168),(122,169),(123,170),(124,171),(125,172),(126,173),(127,174),(128,175),(129,176),(130,177),(131,200),(132,201),(133,202),(134,203),(135,204),(136,205),(137,206),(138,207),(139,208),(140,183),(141,184),(142,185),(143,186),(144,187),(145,188),(146,189),(147,190),(148,191),(149,192),(150,193),(151,194),(152,195),(153,196),(154,197),(155,198),(156,199)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,52),(11,27),(12,28),(13,29),(14,30),(15,31),(16,32),(17,33),(18,34),(19,35),(20,36),(21,37),(22,38),(23,39),(24,40),(25,41),(26,42),(53,83),(54,84),(55,85),(56,86),(57,87),(58,88),(59,89),(60,90),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,97),(68,98),(69,99),(70,100),(71,101),(72,102),(73,103),(74,104),(75,79),(76,80),(77,81),(78,82),(105,131),(106,132),(107,133),(108,134),(109,135),(110,136),(111,137),(112,138),(113,139),(114,140),(115,141),(116,142),(117,143),(118,144),(119,145),(120,146),(121,147),(122,148),(123,149),(124,150),(125,151),(126,152),(127,153),(128,154),(129,155),(130,156),(157,205),(158,206),(159,207),(160,208),(161,183),(162,184),(163,185),(164,186),(165,187),(166,188),(167,189),(168,190),(169,191),(170,192),(171,193),(172,194),(173,195),(174,196),(175,197),(176,198),(177,199),(178,200),(179,201),(180,202),(181,203),(182,204)], [(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···2O13A···13L26A···26FX
order12···213···1326···26
size11···11···11···1

208 irreducible representations

dim1111
type++
imageC1C2C13C26
kernelC23×C26C22×C26C24C23
# reps11512180

Matrix representation of C23×C26 in GL4(𝔽53) generated by

52000
0100
00520
0001
,
1000
0100
0010
00052
,
52000
0100
0010
0001
,
52000
03800
00170
00047
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

׿
×
𝔽