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## G = C22×D52order 416 = 25·13

### Direct product of C22 and D52

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — C22×D52
 Chief series C1 — C13 — C26 — D26 — C22×D13 — C23×D13 — C22×D52
 Lower central C13 — C26 — C22×D52
 Upper central C1 — C23 — C22×C4

Generators and relations for C22×D52
G = < a,b,c,d | a2=b2=c52=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1888 in 236 conjugacy classes, 105 normal (9 characteristic)
C1, C2, C2, C2, C4, C22, C22, C2×C4, D4, C23, C23, C13, C22×C4, C2×D4, C24, D13, C26, C26, C22×D4, C52, D26, D26, C2×C26, D52, C2×C52, C22×D13, C22×D13, C22×C26, C2×D52, C22×C52, C23×D13, C22×D52
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, D13, C22×D4, D26, D52, C22×D13, C2×D52, C23×D13, C22×D52

Smallest permutation representation of C22×D52
On 208 points
Generators in S208
(1 143)(2 144)(3 145)(4 146)(5 147)(6 148)(7 149)(8 150)(9 151)(10 152)(11 153)(12 154)(13 155)(14 156)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 111)(22 112)(23 113)(24 114)(25 115)(26 116)(27 117)(28 118)(29 119)(30 120)(31 121)(32 122)(33 123)(34 124)(35 125)(36 126)(37 127)(38 128)(39 129)(40 130)(41 131)(42 132)(43 133)(44 134)(45 135)(46 136)(47 137)(48 138)(49 139)(50 140)(51 141)(52 142)(53 179)(54 180)(55 181)(56 182)(57 183)(58 184)(59 185)(60 186)(61 187)(62 188)(63 189)(64 190)(65 191)(66 192)(67 193)(68 194)(69 195)(70 196)(71 197)(72 198)(73 199)(74 200)(75 201)(76 202)(77 203)(78 204)(79 205)(80 206)(81 207)(82 208)(83 157)(84 158)(85 159)(86 160)(87 161)(88 162)(89 163)(90 164)(91 165)(92 166)(93 167)(94 168)(95 169)(96 170)(97 171)(98 172)(99 173)(100 174)(101 175)(102 176)(103 177)(104 178)
(1 97)(2 98)(3 99)(4 100)(5 101)(6 102)(7 103)(8 104)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 72)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 81)(38 82)(39 83)(40 84)(41 85)(42 86)(43 87)(44 88)(45 89)(46 90)(47 91)(48 92)(49 93)(50 94)(51 95)(52 96)(105 185)(106 186)(107 187)(108 188)(109 189)(110 190)(111 191)(112 192)(113 193)(114 194)(115 195)(116 196)(117 197)(118 198)(119 199)(120 200)(121 201)(122 202)(123 203)(124 204)(125 205)(126 206)(127 207)(128 208)(129 157)(130 158)(131 159)(132 160)(133 161)(134 162)(135 163)(136 164)(137 165)(138 166)(139 167)(140 168)(141 169)(142 170)(143 171)(144 172)(145 173)(146 174)(147 175)(148 176)(149 177)(150 178)(151 179)(152 180)(153 181)(154 182)(155 183)(156 184)
(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 83)(2 82)(3 81)(4 80)(5 79)(6 78)(7 77)(8 76)(9 75)(10 74)(11 73)(12 72)(13 71)(14 70)(15 69)(16 68)(17 67)(18 66)(19 65)(20 64)(21 63)(22 62)(23 61)(24 60)(25 59)(26 58)(27 57)(28 56)(29 55)(30 54)(31 53)(32 104)(33 103)(34 102)(35 101)(36 100)(37 99)(38 98)(39 97)(40 96)(41 95)(42 94)(43 93)(44 92)(45 91)(46 90)(47 89)(48 88)(49 87)(50 86)(51 85)(52 84)(105 195)(106 194)(107 193)(108 192)(109 191)(110 190)(111 189)(112 188)(113 187)(114 186)(115 185)(116 184)(117 183)(118 182)(119 181)(120 180)(121 179)(122 178)(123 177)(124 176)(125 175)(126 174)(127 173)(128 172)(129 171)(130 170)(131 169)(132 168)(133 167)(134 166)(135 165)(136 164)(137 163)(138 162)(139 161)(140 160)(141 159)(142 158)(143 157)(144 208)(145 207)(146 206)(147 205)(148 204)(149 203)(150 202)(151 201)(152 200)(153 199)(154 198)(155 197)(156 196)

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

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

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

116 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A 4B 4C 4D 13A ··· 13F 26A ··· 26AP 52A ··· 52AV order 1 2 ··· 2 2 ··· 2 4 4 4 4 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 ··· 1 26 ··· 26 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

116 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 D4 D13 D26 D26 D52 kernel C22×D52 C2×D52 C22×C52 C23×D13 C2×C26 C22×C4 C2×C4 C23 C22 # reps 1 12 1 2 4 6 36 6 48

Matrix representation of C22×D52 in GL4(𝔽53) generated by

 52 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 52 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 22 40 0 0 13 2
,
 1 0 0 0 0 1 0 0 0 0 26 14 0 0 1 27
G:=sub<GL(4,GF(53))| [52,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[52,0,0,0,0,52,0,0,0,0,1,0,0,0,0,1],[52,0,0,0,0,1,0,0,0,0,22,13,0,0,40,2],[1,0,0,0,0,1,0,0,0,0,26,1,0,0,14,27] >;

C22×D52 in GAP, Magma, Sage, TeX

C_2^2\times D_{52}
% in TeX

G:=Group("C2^2xD52");
// GroupNames label

G:=SmallGroup(416,214);
// by ID

G=gap.SmallGroup(416,214);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,579,69,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^52=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽