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## G = C2×D52⋊5C2order 416 = 25·13

### Direct product of C2 and D52⋊5C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — C2×D52⋊5C2
 Chief series C1 — C13 — C26 — D26 — C22×D13 — C2×C4×D13 — C2×D52⋊5C2
 Lower central C13 — C26 — C2×D52⋊5C2
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C2×D525C2
G = < a,b,c,d | a2=b52=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd=b26c >

Subgroups: 992 in 164 conjugacy classes, 89 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C4 [×4], C22, C22 [×2], C22 [×10], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×12], Q8 [×4], C23, C23 [×2], C13, C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×8], D13 [×4], C26, C26 [×2], C26 [×2], C2×C4○D4, Dic13 [×4], C52 [×4], D26 [×4], D26 [×4], C2×C26, C2×C26 [×2], C2×C26 [×2], Dic26 [×4], C4×D13 [×8], D52 [×4], C2×Dic13 [×2], C13⋊D4 [×8], C2×C52 [×2], C2×C52 [×4], C22×D13 [×2], C22×C26, C2×Dic26, C2×C4×D13 [×2], C2×D52, D525C2 [×8], C2×C13⋊D4 [×2], C22×C52, C2×D525C2
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×2], C24, D13, C2×C4○D4, D26 [×7], C22×D13 [×7], D525C2 [×2], C23×D13, C2×D525C2

Smallest permutation representation of C2×D525C2
On 208 points
Generators in S208
(1 208)(2 157)(3 158)(4 159)(5 160)(6 161)(7 162)(8 163)(9 164)(10 165)(11 166)(12 167)(13 168)(14 169)(15 170)(16 171)(17 172)(18 173)(19 174)(20 175)(21 176)(22 177)(23 178)(24 179)(25 180)(26 181)(27 182)(28 183)(29 184)(30 185)(31 186)(32 187)(33 188)(34 189)(35 190)(36 191)(37 192)(38 193)(39 194)(40 195)(41 196)(42 197)(43 198)(44 199)(45 200)(46 201)(47 202)(48 203)(49 204)(50 205)(51 206)(52 207)(53 140)(54 141)(55 142)(56 143)(57 144)(58 145)(59 146)(60 147)(61 148)(62 149)(63 150)(64 151)(65 152)(66 153)(67 154)(68 155)(69 156)(70 105)(71 106)(72 107)(73 108)(74 109)(75 110)(76 111)(77 112)(78 113)(79 114)(80 115)(81 116)(82 117)(83 118)(84 119)(85 120)(86 121)(87 122)(88 123)(89 124)(90 125)(91 126)(92 127)(93 128)(94 129)(95 130)(96 131)(97 132)(98 133)(99 134)(100 135)(101 136)(102 137)(103 138)(104 139)
(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 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(53 90)(54 89)(55 88)(56 87)(57 86)(58 85)(59 84)(60 83)(61 82)(62 81)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(91 104)(92 103)(93 102)(94 101)(95 100)(96 99)(97 98)(105 108)(106 107)(109 156)(110 155)(111 154)(112 153)(113 152)(114 151)(115 150)(116 149)(117 148)(118 147)(119 146)(120 145)(121 144)(122 143)(123 142)(124 141)(125 140)(126 139)(127 138)(128 137)(129 136)(130 135)(131 134)(132 133)(157 206)(158 205)(159 204)(160 203)(161 202)(162 201)(163 200)(164 199)(165 198)(166 197)(167 196)(168 195)(169 194)(170 193)(171 192)(172 191)(173 190)(174 189)(175 188)(176 187)(177 186)(178 185)(179 184)(180 183)(181 182)(207 208)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 79)(22 80)(23 81)(24 82)(25 83)(26 84)(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 53)(48 54)(49 55)(50 56)(51 57)(52 58)(105 167)(106 168)(107 169)(108 170)(109 171)(110 172)(111 173)(112 174)(113 175)(114 176)(115 177)(116 178)(117 179)(118 180)(119 181)(120 182)(121 183)(122 184)(123 185)(124 186)(125 187)(126 188)(127 189)(128 190)(129 191)(130 192)(131 193)(132 194)(133 195)(134 196)(135 197)(136 198)(137 199)(138 200)(139 201)(140 202)(141 203)(142 204)(143 205)(144 206)(145 207)(146 208)(147 157)(148 158)(149 159)(150 160)(151 161)(152 162)(153 163)(154 164)(155 165)(156 166)

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

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

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

116 conjugacy classes

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

116 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4○D4 D13 D26 D26 D52⋊5C2 kernel C2×D52⋊5C2 C2×Dic26 C2×C4×D13 C2×D52 D52⋊5C2 C2×C13⋊D4 C22×C52 C26 C22×C4 C2×C4 C23 C2 # reps 1 1 2 1 8 2 1 4 6 36 6 48

Matrix representation of C2×D525C2 in GL3(𝔽53) generated by

 52 0 0 0 1 0 0 0 1
,
 1 0 0 0 50 38 0 31 31
,
 52 0 0 0 45 32 0 3 8
,
 52 0 0 0 6 24 0 14 47
G:=sub<GL(3,GF(53))| [52,0,0,0,1,0,0,0,1],[1,0,0,0,50,31,0,38,31],[52,0,0,0,45,3,0,32,8],[52,0,0,0,6,14,0,24,47] >;

C2×D525C2 in GAP, Magma, Sage, TeX

C_2\times D_{52}\rtimes_5C_2
% in TeX

G:=Group("C2xD52:5C2");
// GroupNames label

G:=SmallGroup(416,215);
// by ID

G=gap.SmallGroup(416,215);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,86,579,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^52=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d=b^26*c>;
// generators/relations

׿
×
𝔽