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## G = C2×D52⋊C2order 416 = 25·13

### Direct product of C2 and D52⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — C2×D52⋊C2
 Chief series C1 — C13 — C26 — D26 — C22×D13 — C2×C4×D13 — C2×D52⋊C2
 Lower central C13 — C26 — C2×D52⋊C2
 Upper central C1 — C22 — C2×Q8

Generators and relations for C2×D52⋊C2
G = < a,b,c,d | a2=b52=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd=b25, dcd=b50c >

Subgroups: 1088 in 164 conjugacy classes, 89 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×6], C4 [×2], C22, C22 [×12], C2×C4 [×3], C2×C4 [×13], D4 [×12], Q8 [×4], C23 [×3], C13, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], D13 [×6], C26, C26 [×2], C2×C4○D4, Dic13 [×2], C52 [×6], D26 [×6], D26 [×6], C2×C26, C4×D13 [×12], D52 [×12], C2×Dic13, C2×C52 [×3], Q8×C13 [×4], C22×D13 [×3], C2×C4×D13 [×3], C2×D52 [×3], D52⋊C2 [×8], Q8×C26, C2×D52⋊C2
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×2], C24, D13, C2×C4○D4, D26 [×7], C22×D13 [×7], D52⋊C2 [×2], C23×D13, C2×D52⋊C2

Smallest permutation representation of C2×D52⋊C2
On 208 points
Generators in S208
(1 155)(2 156)(3 105)(4 106)(5 107)(6 108)(7 109)(8 110)(9 111)(10 112)(11 113)(12 114)(13 115)(14 116)(15 117)(16 118)(17 119)(18 120)(19 121)(20 122)(21 123)(22 124)(23 125)(24 126)(25 127)(26 128)(27 129)(28 130)(29 131)(30 132)(31 133)(32 134)(33 135)(34 136)(35 137)(36 138)(37 139)(38 140)(39 141)(40 142)(41 143)(42 144)(43 145)(44 146)(45 147)(46 148)(47 149)(48 150)(49 151)(50 152)(51 153)(52 154)(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 63)(2 62)(3 61)(4 60)(5 59)(6 58)(7 57)(8 56)(9 55)(10 54)(11 53)(12 104)(13 103)(14 102)(15 101)(16 100)(17 99)(18 98)(19 97)(20 96)(21 95)(22 94)(23 93)(24 92)(25 91)(26 90)(27 89)(28 88)(29 87)(30 86)(31 85)(32 84)(33 83)(34 82)(35 81)(36 80)(37 79)(38 78)(39 77)(40 76)(41 75)(42 74)(43 73)(44 72)(45 71)(46 70)(47 69)(48 68)(49 67)(50 66)(51 65)(52 64)(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)
(2 26)(3 51)(4 24)(5 49)(6 22)(7 47)(8 20)(9 45)(10 18)(11 43)(12 16)(13 41)(15 39)(17 37)(19 35)(21 33)(23 31)(25 29)(28 52)(30 50)(32 48)(34 46)(36 44)(38 42)(53 75)(54 100)(55 73)(56 98)(57 71)(58 96)(59 69)(60 94)(61 67)(62 92)(63 65)(64 90)(66 88)(68 86)(70 84)(72 82)(74 80)(76 78)(77 103)(79 101)(81 99)(83 97)(85 95)(87 93)(89 91)(102 104)(105 153)(106 126)(107 151)(108 124)(109 149)(110 122)(111 147)(112 120)(113 145)(114 118)(115 143)(117 141)(119 139)(121 137)(123 135)(125 133)(127 131)(128 156)(130 154)(132 152)(134 150)(136 148)(138 146)(140 144)(157 187)(158 160)(159 185)(161 183)(162 208)(163 181)(164 206)(165 179)(166 204)(167 177)(168 202)(169 175)(170 200)(171 173)(172 198)(174 196)(176 194)(178 192)(180 190)(182 188)(184 186)(189 207)(191 205)(193 203)(195 201)(197 199)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A ··· 4F 4G 4H 4I 4J 13A ··· 13F 26A ··· 26R 52A ··· 52AJ order 1 2 2 2 2 ··· 2 4 ··· 4 4 4 4 4 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 1 1 26 ··· 26 2 ··· 2 13 13 13 13 2 ··· 2 2 ··· 2 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C4○D4 D13 D26 D26 D52⋊C2 kernel C2×D52⋊C2 C2×C4×D13 C2×D52 D52⋊C2 Q8×C26 C26 C2×Q8 C2×C4 Q8 C2 # reps 1 3 3 8 1 4 6 18 24 12

Matrix representation of C2×D52⋊C2 in GL4(𝔽53) generated by

 52 0 0 0 0 52 0 0 0 0 1 0 0 0 0 1
,
 9 9 0 0 0 6 0 0 0 0 30 0 0 0 49 23
,
 36 24 0 0 41 17 0 0 0 0 5 22 0 0 23 48
,
 49 2 0 0 19 4 0 0 0 0 1 0 0 0 14 52
G:=sub<GL(4,GF(53))| [52,0,0,0,0,52,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,9,6,0,0,0,0,30,49,0,0,0,23],[36,41,0,0,24,17,0,0,0,0,5,23,0,0,22,48],[49,19,0,0,2,4,0,0,0,0,1,14,0,0,0,52] >;

C2×D52⋊C2 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(416,220);
// by ID

G=gap.SmallGroup(416,220);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,86,579,159,69,13829]);
// 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,d*b*d=b^25,d*c*d=b^50*c>;
// generators/relations

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