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## G = C22×C52order 208 = 24·13

### Abelian group of type [2,2,52]

Aliases: C22×C52, SmallGroup(208,45)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C52
 Chief series C1 — C2 — C26 — C52 — C2×C52 — C22×C52
 Lower central C1 — C22×C52
 Upper central C1 — C22×C52

Generators and relations for C22×C52
G = < a,b,c | a2=b2=c52=1, ab=ba, ac=ca, bc=cb >

Subgroups: 54, all normal (8 characteristic)
C1, C2, C2 [×6], C4 [×4], C22 [×7], C2×C4 [×6], C23, C13, C22×C4, C26, C26 [×6], C52 [×4], C2×C26 [×7], C2×C52 [×6], C22×C26, C22×C52
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C13, C22×C4, C26 [×7], C52 [×4], C2×C26 [×7], C2×C52 [×6], C22×C26, C22×C52

Smallest permutation representation of C22×C52
Regular action on 208 points
Generators in S208
(1 111)(2 112)(3 113)(4 114)(5 115)(6 116)(7 117)(8 118)(9 119)(10 120)(11 121)(12 122)(13 123)(14 124)(15 125)(16 126)(17 127)(18 128)(19 129)(20 130)(21 131)(22 132)(23 133)(24 134)(25 135)(26 136)(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)(47 105)(48 106)(49 107)(50 108)(51 109)(52 110)(53 202)(54 203)(55 204)(56 205)(57 206)(58 207)(59 208)(60 157)(61 158)(62 159)(63 160)(64 161)(65 162)(66 163)(67 164)(68 165)(69 166)(70 167)(71 168)(72 169)(73 170)(74 171)(75 172)(76 173)(77 174)(78 175)(79 176)(80 177)(81 178)(82 179)(83 180)(84 181)(85 182)(86 183)(87 184)(88 185)(89 186)(90 187)(91 188)(92 189)(93 190)(94 191)(95 192)(96 193)(97 194)(98 195)(99 196)(100 197)(101 198)(102 199)(103 200)(104 201)
(1 82)(2 83)(3 84)(4 85)(5 86)(6 87)(7 88)(8 89)(9 90)(10 91)(11 92)(12 93)(13 94)(14 95)(15 96)(16 97)(17 98)(18 99)(19 100)(20 101)(21 102)(22 103)(23 104)(24 53)(25 54)(26 55)(27 56)(28 57)(29 58)(30 59)(31 60)(32 61)(33 62)(34 63)(35 64)(36 65)(37 66)(38 67)(39 68)(40 69)(41 70)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)(51 80)(52 81)(105 173)(106 174)(107 175)(108 176)(109 177)(110 178)(111 179)(112 180)(113 181)(114 182)(115 183)(116 184)(117 185)(118 186)(119 187)(120 188)(121 189)(122 190)(123 191)(124 192)(125 193)(126 194)(127 195)(128 196)(129 197)(130 198)(131 199)(132 200)(133 201)(134 202)(135 203)(136 204)(137 205)(138 206)(139 207)(140 208)(141 157)(142 158)(143 159)(144 160)(145 161)(146 162)(147 163)(148 164)(149 165)(150 166)(151 167)(152 168)(153 169)(154 170)(155 171)(156 172)
(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)| (1,111)(2,112)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,119)(10,120)(11,121)(12,122)(13,123)(14,124)(15,125)(16,126)(17,127)(18,128)(19,129)(20,130)(21,131)(22,132)(23,133)(24,134)(25,135)(26,136)(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)(47,105)(48,106)(49,107)(50,108)(51,109)(52,110)(53,202)(54,203)(55,204)(56,205)(57,206)(58,207)(59,208)(60,157)(61,158)(62,159)(63,160)(64,161)(65,162)(66,163)(67,164)(68,165)(69,166)(70,167)(71,168)(72,169)(73,170)(74,171)(75,172)(76,173)(77,174)(78,175)(79,176)(80,177)(81,178)(82,179)(83,180)(84,181)(85,182)(86,183)(87,184)(88,185)(89,186)(90,187)(91,188)(92,189)(93,190)(94,191)(95,192)(96,193)(97,194)(98,195)(99,196)(100,197)(101,198)(102,199)(103,200)(104,201), (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,101)(21,102)(22,103)(23,104)(24,53)(25,54)(26,55)(27,56)(28,57)(29,58)(30,59)(31,60)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,81)(105,173)(106,174)(107,175)(108,176)(109,177)(110,178)(111,179)(112,180)(113,181)(114,182)(115,183)(116,184)(117,185)(118,186)(119,187)(120,188)(121,189)(122,190)(123,191)(124,192)(125,193)(126,194)(127,195)(128,196)(129,197)(130,198)(131,199)(132,200)(133,201)(134,202)(135,203)(136,204)(137,205)(138,206)(139,207)(140,208)(141,157)(142,158)(143,159)(144,160)(145,161)(146,162)(147,163)(148,164)(149,165)(150,166)(151,167)(152,168)(153,169)(154,170)(155,171)(156,172), (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( (1,111)(2,112)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,119)(10,120)(11,121)(12,122)(13,123)(14,124)(15,125)(16,126)(17,127)(18,128)(19,129)(20,130)(21,131)(22,132)(23,133)(24,134)(25,135)(26,136)(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)(47,105)(48,106)(49,107)(50,108)(51,109)(52,110)(53,202)(54,203)(55,204)(56,205)(57,206)(58,207)(59,208)(60,157)(61,158)(62,159)(63,160)(64,161)(65,162)(66,163)(67,164)(68,165)(69,166)(70,167)(71,168)(72,169)(73,170)(74,171)(75,172)(76,173)(77,174)(78,175)(79,176)(80,177)(81,178)(82,179)(83,180)(84,181)(85,182)(86,183)(87,184)(88,185)(89,186)(90,187)(91,188)(92,189)(93,190)(94,191)(95,192)(96,193)(97,194)(98,195)(99,196)(100,197)(101,198)(102,199)(103,200)(104,201), (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,101)(21,102)(22,103)(23,104)(24,53)(25,54)(26,55)(27,56)(28,57)(29,58)(30,59)(31,60)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,81)(105,173)(106,174)(107,175)(108,176)(109,177)(110,178)(111,179)(112,180)(113,181)(114,182)(115,183)(116,184)(117,185)(118,186)(119,187)(120,188)(121,189)(122,190)(123,191)(124,192)(125,193)(126,194)(127,195)(128,196)(129,197)(130,198)(131,199)(132,200)(133,201)(134,202)(135,203)(136,204)(137,205)(138,206)(139,207)(140,208)(141,157)(142,158)(143,159)(144,160)(145,161)(146,162)(147,163)(148,164)(149,165)(150,166)(151,167)(152,168)(153,169)(154,170)(155,171)(156,172), (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([(1,111),(2,112),(3,113),(4,114),(5,115),(6,116),(7,117),(8,118),(9,119),(10,120),(11,121),(12,122),(13,123),(14,124),(15,125),(16,126),(17,127),(18,128),(19,129),(20,130),(21,131),(22,132),(23,133),(24,134),(25,135),(26,136),(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),(47,105),(48,106),(49,107),(50,108),(51,109),(52,110),(53,202),(54,203),(55,204),(56,205),(57,206),(58,207),(59,208),(60,157),(61,158),(62,159),(63,160),(64,161),(65,162),(66,163),(67,164),(68,165),(69,166),(70,167),(71,168),(72,169),(73,170),(74,171),(75,172),(76,173),(77,174),(78,175),(79,176),(80,177),(81,178),(82,179),(83,180),(84,181),(85,182),(86,183),(87,184),(88,185),(89,186),(90,187),(91,188),(92,189),(93,190),(94,191),(95,192),(96,193),(97,194),(98,195),(99,196),(100,197),(101,198),(102,199),(103,200),(104,201)], [(1,82),(2,83),(3,84),(4,85),(5,86),(6,87),(7,88),(8,89),(9,90),(10,91),(11,92),(12,93),(13,94),(14,95),(15,96),(16,97),(17,98),(18,99),(19,100),(20,101),(21,102),(22,103),(23,104),(24,53),(25,54),(26,55),(27,56),(28,57),(29,58),(30,59),(31,60),(32,61),(33,62),(34,63),(35,64),(36,65),(37,66),(38,67),(39,68),(40,69),(41,70),(42,71),(43,72),(44,73),(45,74),(46,75),(47,76),(48,77),(49,78),(50,79),(51,80),(52,81),(105,173),(106,174),(107,175),(108,176),(109,177),(110,178),(111,179),(112,180),(113,181),(114,182),(115,183),(116,184),(117,185),(118,186),(119,187),(120,188),(121,189),(122,190),(123,191),(124,192),(125,193),(126,194),(127,195),(128,196),(129,197),(130,198),(131,199),(132,200),(133,201),(134,202),(135,203),(136,204),(137,205),(138,206),(139,207),(140,208),(141,157),(142,158),(143,159),(144,160),(145,161),(146,162),(147,163),(148,164),(149,165),(150,166),(151,167),(152,168),(153,169),(154,170),(155,171),(156,172)], [(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)])

C22×C52 is a maximal subgroup of   C52.55D4  C26.10C42  C52.48D4  C23.21D26  C23.23D26  C527D4

208 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 13A ··· 13L 26A ··· 26CF 52A ··· 52CR order 1 2 ··· 2 4 ··· 4 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

208 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C4 C13 C26 C26 C52 kernel C22×C52 C2×C52 C22×C26 C2×C26 C22×C4 C2×C4 C23 C22 # reps 1 6 1 8 12 72 12 96

Matrix representation of C22×C52 in GL3(𝔽53) generated by

 52 0 0 0 52 0 0 0 52
,
 1 0 0 0 52 0 0 0 1
,
 13 0 0 0 18 0 0 0 34
G:=sub<GL(3,GF(53))| [52,0,0,0,52,0,0,0,52],[1,0,0,0,52,0,0,0,1],[13,0,0,0,18,0,0,0,34] >;

C22×C52 in GAP, Magma, Sage, TeX

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

G:=Group("C2^2xC52");
// GroupNames label

G:=SmallGroup(208,45);
// by ID

G=gap.SmallGroup(208,45);
# by ID

G:=PCGroup([5,-2,-2,-2,-13,-2,520]);
// Polycyclic

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

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