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G = C22×D52order 416 = 25·13

Direct product of C22 and D52

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22×D52, C522C23, D261C23, C26.3C24, C23.35D26, (C2×C26)⋊6D4, C261(C2×D4), (C2×C4)⋊9D26, C131(C22×D4), (C22×C52)⋊7C2, C42(C22×D13), (C22×C4)⋊5D13, (C2×C52)⋊12C22, (C23×D13)⋊3C2, C2.4(C23×D13), (C2×C26).64C23, (C22×D13)⋊5C22, (C22×C26).45C22, C22.30(C22×D13), SmallGroup(416,214)

Series: Derived Chief Lower central Upper central

C1C26 — C22×D52
C1C13C26D26C22×D13C23×D13 — C22×D52
C13C26 — C22×D52
C1C23C22×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 [×6], C2 [×8], C4 [×4], C22 [×7], C22 [×32], C2×C4 [×6], D4 [×16], C23, C23 [×20], C13, C22×C4, C2×D4 [×12], C24 [×2], D13 [×8], C26, C26 [×6], C22×D4, C52 [×4], D26 [×8], D26 [×24], C2×C26 [×7], D52 [×16], C2×C52 [×6], C22×D13 [×12], C22×D13 [×8], C22×C26, C2×D52 [×12], C22×C52, C23×D13 [×2], C22×D52
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, D13, C22×D4, D26 [×7], D52 [×4], C22×D13 [×7], C2×D52 [×6], C23×D13, C22×D52

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

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

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

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

116 conjugacy classes

class 1 2A···2G2H···2O4A4B4C4D13A···13F26A···26AP52A···52AV
order12···22···2444413···1326···2652···52
size11···126···2622222···22···22···2

116 irreducible representations

dim111122222
type+++++++++
imageC1C2C2C2D4D13D26D26D52
kernelC22×D52C2×D52C22×C52C23×D13C2×C26C22×C4C2×C4C23C22
# reps112124636648

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

52000
0100
0010
0001
,
52000
05200
0010
0001
,
52000
0100
002240
00132
,
1000
0100
002614
00127
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