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## G = C22×Dic13order 208 = 24·13

### Direct product of C22 and Dic13

Aliases: C22×Dic13, C26.9C23, C23.2D13, C22.11D26, C263(C2×C4), (C2×C26)⋊5C4, C133(C22×C4), (C22×C26).3C2, C2.2(C22×D13), (C2×C26).12C22, SmallGroup(208,43)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C22×Dic13
 Chief series C1 — C13 — C26 — Dic13 — C2×Dic13 — C22×Dic13
 Lower central C13 — C22×Dic13
 Upper central C1 — C23

Generators and relations for C22×Dic13
G = < a,b,c,d | a2=b2=c26=1, d2=c13, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 186 in 54 conjugacy classes, 43 normal (7 characteristic)
C1, C2, C2, C4, C22, C2×C4, C23, C13, C22×C4, C26, C26, Dic13, C2×C26, C2×Dic13, C22×C26, C22×Dic13
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, D13, Dic13, D26, C2×Dic13, C22×D13, C22×Dic13

Smallest permutation representation of C22×Dic13
Regular action on 208 points
Generators in S208
(1 60)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 67)(9 68)(10 69)(11 70)(12 71)(13 72)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 53)(21 54)(22 55)(23 56)(24 57)(25 58)(26 59)(27 85)(28 86)(29 87)(30 88)(31 89)(32 90)(33 91)(34 92)(35 93)(36 94)(37 95)(38 96)(39 97)(40 98)(41 99)(42 100)(43 101)(44 102)(45 103)(46 104)(47 79)(48 80)(49 81)(50 82)(51 83)(52 84)(105 157)(106 158)(107 159)(108 160)(109 161)(110 162)(111 163)(112 164)(113 165)(114 166)(115 167)(116 168)(117 169)(118 170)(119 171)(120 172)(121 173)(122 174)(123 175)(124 176)(125 177)(126 178)(127 179)(128 180)(129 181)(130 182)(131 196)(132 197)(133 198)(134 199)(135 200)(136 201)(137 202)(138 203)(139 204)(140 205)(141 206)(142 207)(143 208)(144 183)(145 184)(146 185)(147 186)(148 187)(149 188)(150 189)(151 190)(152 191)(153 192)(154 193)(155 194)(156 195)
(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 92)(54 93)(55 94)(56 95)(57 96)(58 97)(59 98)(60 99)(61 100)(62 101)(63 102)(64 103)(65 104)(66 79)(67 80)(68 81)(69 82)(70 83)(71 84)(72 85)(73 86)(74 87)(75 88)(76 89)(77 90)(78 91)(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)(157 196)(158 197)(159 198)(160 199)(161 200)(162 201)(163 202)(164 203)(165 204)(166 205)(167 206)(168 207)(169 208)(170 183)(171 184)(172 185)(173 186)(174 187)(175 188)(176 189)(177 190)(178 191)(179 192)(180 193)(181 194)(182 195)
(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)
(1 131 14 144)(2 156 15 143)(3 155 16 142)(4 154 17 141)(5 153 18 140)(6 152 19 139)(7 151 20 138)(8 150 21 137)(9 149 22 136)(10 148 23 135)(11 147 24 134)(12 146 25 133)(13 145 26 132)(27 119 40 106)(28 118 41 105)(29 117 42 130)(30 116 43 129)(31 115 44 128)(32 114 45 127)(33 113 46 126)(34 112 47 125)(35 111 48 124)(36 110 49 123)(37 109 50 122)(38 108 51 121)(39 107 52 120)(53 203 66 190)(54 202 67 189)(55 201 68 188)(56 200 69 187)(57 199 70 186)(58 198 71 185)(59 197 72 184)(60 196 73 183)(61 195 74 208)(62 194 75 207)(63 193 76 206)(64 192 77 205)(65 191 78 204)(79 177 92 164)(80 176 93 163)(81 175 94 162)(82 174 95 161)(83 173 96 160)(84 172 97 159)(85 171 98 158)(86 170 99 157)(87 169 100 182)(88 168 101 181)(89 167 102 180)(90 166 103 179)(91 165 104 178)

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

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

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

C22×Dic13 is a maximal subgroup of
C26.10C42  C26.M4(2)  C23.11D26  C22⋊Dic26  Dic134D4  C22.D52  C23.18D26  Dic13⋊D4  C22×C4×D13
C22×Dic13 is a maximal quotient of
C23.21D26  D4.Dic13

64 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 13A ··· 13F 26A ··· 26AP order 1 2 ··· 2 4 ··· 4 13 ··· 13 26 ··· 26 size 1 1 ··· 1 13 ··· 13 2 ··· 2 2 ··· 2

64 irreducible representations

 dim 1 1 1 1 2 2 2 type + + + + - + image C1 C2 C2 C4 D13 Dic13 D26 kernel C22×Dic13 C2×Dic13 C22×C26 C2×C26 C23 C22 C22 # reps 1 6 1 8 6 24 18

Matrix representation of C22×Dic13 in GL4(𝔽53) generated by

 1 0 0 0 0 52 0 0 0 0 1 0 0 0 0 1
,
 52 0 0 0 0 1 0 0 0 0 52 0 0 0 0 52
,
 1 0 0 0 0 1 0 0 0 0 0 52 0 0 1 15
,
 52 0 0 0 0 1 0 0 0 0 15 44 0 0 31 38
G:=sub<GL(4,GF(53))| [1,0,0,0,0,52,0,0,0,0,1,0,0,0,0,1],[52,0,0,0,0,1,0,0,0,0,52,0,0,0,0,52],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,52,15],[52,0,0,0,0,1,0,0,0,0,15,31,0,0,44,38] >;

C22×Dic13 in GAP, Magma, Sage, TeX

C_2^2\times {\rm Dic}_{13}
% in TeX

G:=Group("C2^2xDic13");
// GroupNames label

G:=SmallGroup(208,43);
// by ID

G=gap.SmallGroup(208,43);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,40,4804]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^26=1,d^2=c^13,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽