C2^2:Dic26 - GroupNames

G = C₂²⋊Dic₂₆ order 416 = 2⁵·13

The semidirect product of C₂² and Dic₂₆ acting via Dic₂₆/Dic₁₃=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂²⋊₁Dic₂₆, C₂³.₁₂D₂₆, Dic₁₃.₁₄D₄, (C₂×C₂₆)⋊Q₈, C₅₂⋊₃C₄⋊₂C₂, (C₂×C₄).₅D₂₆, C₂.₆(D₄×D₁₃), C₂₆.₄(C₂×Q₈), C₂₆.₁₆(C₂×D₄), C₁₃⋊₁(C₂²⋊Q₈), (C₂×Dic₂₆)⋊₂C₂, C₂₆.D₄⋊₄C₂, (C₂×C₅₂).₁C₂², C₂²⋊C₄.₁D₁₃, C₂.₆(C₂×Dic₂₆), C₂₆.₂₁(C₄○D₄), (C₂×C₂₆).₁₉C₂³, C₂³.D₁₃.₂C₂, C₂.₆(D₄⋊₂D₁₃), (C₂²×C₂₆).₈C₂², (C₂×Dic₁₃).₅C₂², (C₂²×Dic₁₃).₃C₂, C₂².₃₉(C₂²×D₁₃), (C₁₃×C₂²⋊C₄).₁C₂, SmallGroup(416,99)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₆ — C₂²⋊Dic₂₆

Chief series

C₁ — C₁₃ — C₂₆ — C₂×C₂₆ — C₂×Dic₁₃ — C₂²×Dic₁₃ — C₂²⋊Dic₂₆

Lower central

C₁₃ — C₂×C₂₆ — C₂²⋊Dic₂₆

Upper central

C₁ — C₂² — C₂²⋊C₄

Generators and relations for C₂²⋊Dic₂₆
G = < a,b,c,d | a²=b²=c⁵²=1, d²=c²⁶, cac^-1=ab=ba, ad=da, bc=cb, bd=db, dcd^-1=c^-1 >

Subgroups: 424 in 74 conjugacy classes, 35 normal (29 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₁₃, C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×Q₈, C₂₆, C₂₆, C₂²⋊Q₈, Dic₁₃, Dic₁₃, C₅₂, C₂×C₂₆, C₂×C₂₆, C₂×C₂₆, Dic₂₆, C₂×Dic₁₃, C₂×Dic₁₃, C₂×C₅₂, C₂²×C₂₆, C₂₆.D₄, C₅₂⋊₃C₄, C₂³.D₁₃, C₁₃×C₂²⋊C₄, C₂×Dic₂₆, C₂²×Dic₁₃, C₂²⋊Dic₂₆
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, D₁₃, C₂²⋊Q₈, D₂₆, Dic₂₆, C₂²×D₁₃, C₂×Dic₂₆, D₄×D₁₃, D₄⋊₂D₁₃, C₂²⋊Dic₂₆

Smallest permutation representation of C₂²⋊Dic₂₆
►On 208 points

Generators in S₂₀₈

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

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

(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)

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

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

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

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

74 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	13A	···	13F	26A	···	26R	26S	···	26AD	52A	···	52X
order	1	2	2	2	2	2	4	4	4	4	4	4	4	4	13	···	13	26	···	26	26	···	26	52	···	52
size	1	1	1	1	2	2	4	4	26	26	26	26	52	52	2	···	2	2	···	2	4	···	4	4	···	4

74 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+	+	+	-		+	+	+	-	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	D₄	Q₈	C₄○D₄	D₁₃	D₂₆	D₂₆	Dic₂₆	D₄×D₁₃	D₄⋊₂D₁₃
kernel	C₂²⋊Dic₂₆	C₂₆.D₄	C₅₂⋊₃C₄	C₂³.D₁₃	C₁₃×C₂²⋊C₄	C₂×Dic₂₆	C₂²×Dic₁₃	Dic₁₃	C₂×C₂₆	C₂₆	C₂²⋊C₄	C₂×C₄	C₂³	C₂²	C₂	C₂
# reps	1	2	1	1	1	1	1	2	2	2	6	12	6	24	6	6

Matrix representation of C₂²⋊Dic₂₆ ►in GL₆(𝔽₅₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	52	0	0	0
0	0	0	52	0	0
0	0	0	0	52	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	52	0
0	0	0	0	0	52

9	52	0	0	0	0
51	18	0	0	0	0
0	0	6	8	0	0
0	0	2	47	0	0
0	0	0	0	0	1
0	0	0	0	52	0

45	10	0	0	0	0
52	8	0	0	0	0
0	0	21	5	0	0
0	0	7	32	0	0
0	0	0	0	23	0
0	0	0	0	0	30

G:=sub<GL(6,GF(53))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,52,0,0,0,0,0,0,52,0,0,0,0,0,0,52,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,52,0,0,0,0,0,0,52],[9,51,0,0,0,0,52,18,0,0,0,0,0,0,6,2,0,0,0,0,8,47,0,0,0,0,0,0,0,52,0,0,0,0,1,0],[45,52,0,0,0,0,10,8,0,0,0,0,0,0,21,7,0,0,0,0,5,32,0,0,0,0,0,0,23,0,0,0,0,0,0,30] >;

C₂²⋊Dic₂₆ in GAP, Magma, Sage, TeX

C_2^2\rtimes {\rm Dic}_{26}

% in TeX

G:=Group("C2^2:Dic26");

// GroupNames label

G:=SmallGroup(416,99);

// by ID

G=gap.SmallGroup(416,99);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,218,188,50,13829]);

// Polycyclic

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

// generators/relations

G = C22⋊Dic26 order 416 = 25·13

The semidirect product of C22 and Dic26 acting via Dic26/Dic13=C2

G = C₂²⋊Dic₂₆ order 416 = 2⁵·13

The semidirect product of C₂² and Dic₂₆ acting via Dic₂₆/Dic₁₃=C₂