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## G = D4×C2×C26order 416 = 25·13

### Direct product of C2×C26 and D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C2×C26
 Chief series C1 — C2 — C26 — C2×C26 — D4×C13 — D4×C26 — D4×C2×C26
 Lower central C1 — C2 — D4×C2×C26
 Upper central C1 — C22×C26 — D4×C2×C26

Generators and relations for D4×C2×C26
G = < a,b,c,d | a2=b26=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 316 in 236 conjugacy classes, 156 normal (10 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×4], C22 [×15], C22 [×24], C2×C4 [×6], D4 [×16], C23, C23 [×12], C23 [×8], C13, C22×C4, C2×D4 [×12], C24 [×2], C26, C26 [×6], C26 [×8], C22×D4, C52 [×4], C2×C26 [×15], C2×C26 [×24], C2×C52 [×6], D4×C13 [×16], C22×C26, C22×C26 [×12], C22×C26 [×8], C22×C52, D4×C26 [×12], C23×C26 [×2], D4×C2×C26
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C13, C2×D4 [×6], C24, C26 [×15], C22×D4, C2×C26 [×35], D4×C13 [×4], C22×C26 [×15], D4×C26 [×6], C23×C26, D4×C2×C26

Smallest permutation representation of D4×C2×C26
On 208 points
Generators in S208
(1 179)(2 180)(3 181)(4 182)(5 157)(6 158)(7 159)(8 160)(9 161)(10 162)(11 163)(12 164)(13 165)(14 166)(15 167)(16 168)(17 169)(18 170)(19 171)(20 172)(21 173)(22 174)(23 175)(24 176)(25 177)(26 178)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 53)(50 54)(51 55)(52 56)(79 142)(80 143)(81 144)(82 145)(83 146)(84 147)(85 148)(86 149)(87 150)(88 151)(89 152)(90 153)(91 154)(92 155)(93 156)(94 131)(95 132)(96 133)(97 134)(98 135)(99 136)(100 137)(101 138)(102 139)(103 140)(104 141)(105 197)(106 198)(107 199)(108 200)(109 201)(110 202)(111 203)(112 204)(113 205)(114 206)(115 207)(116 208)(117 183)(118 184)(119 185)(120 186)(121 187)(122 188)(123 189)(124 190)(125 191)(126 192)(127 193)(128 194)(129 195)(130 196)
(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 76 143 124)(2 77 144 125)(3 78 145 126)(4 53 146 127)(5 54 147 128)(6 55 148 129)(7 56 149 130)(8 57 150 105)(9 58 151 106)(10 59 152 107)(11 60 153 108)(12 61 154 109)(13 62 155 110)(14 63 156 111)(15 64 131 112)(16 65 132 113)(17 66 133 114)(18 67 134 115)(19 68 135 116)(20 69 136 117)(21 70 137 118)(22 71 138 119)(23 72 139 120)(24 73 140 121)(25 74 141 122)(26 75 142 123)(27 87 197 160)(28 88 198 161)(29 89 199 162)(30 90 200 163)(31 91 201 164)(32 92 202 165)(33 93 203 166)(34 94 204 167)(35 95 205 168)(36 96 206 169)(37 97 207 170)(38 98 208 171)(39 99 183 172)(40 100 184 173)(41 101 185 174)(42 102 186 175)(43 103 187 176)(44 104 188 177)(45 79 189 178)(46 80 190 179)(47 81 191 180)(48 82 192 181)(49 83 193 182)(50 84 194 157)(51 85 195 158)(52 86 196 159)
(1 93)(2 94)(3 95)(4 96)(5 97)(6 98)(7 99)(8 100)(9 101)(10 102)(11 103)(12 104)(13 79)(14 80)(15 81)(16 82)(17 83)(18 84)(19 85)(20 86)(21 87)(22 88)(23 89)(24 90)(25 91)(26 92)(27 70)(28 71)(29 72)(30 73)(31 74)(32 75)(33 76)(34 77)(35 78)(36 53)(37 54)(38 55)(39 56)(40 57)(41 58)(42 59)(43 60)(44 61)(45 62)(46 63)(47 64)(48 65)(49 66)(50 67)(51 68)(52 69)(105 184)(106 185)(107 186)(108 187)(109 188)(110 189)(111 190)(112 191)(113 192)(114 193)(115 194)(116 195)(117 196)(118 197)(119 198)(120 199)(121 200)(122 201)(123 202)(124 203)(125 204)(126 205)(127 206)(128 207)(129 208)(130 183)(131 180)(132 181)(133 182)(134 157)(135 158)(136 159)(137 160)(138 161)(139 162)(140 163)(141 164)(142 165)(143 166)(144 167)(145 168)(146 169)(147 170)(148 171)(149 172)(150 173)(151 174)(152 175)(153 176)(154 177)(155 178)(156 179)

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

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

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

260 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A 4B 4C 4D 13A ··· 13L 26A ··· 26CF 26CG ··· 26FX 52A ··· 52AV order 1 2 ··· 2 2 ··· 2 4 4 4 4 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 size 1 1 ··· 1 2 ··· 2 2 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

260 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C13 C26 C26 C26 D4 D4×C13 kernel D4×C2×C26 C22×C52 D4×C26 C23×C26 C22×D4 C22×C4 C2×D4 C24 C2×C26 C22 # reps 1 1 12 2 12 12 144 24 4 48

Matrix representation of D4×C2×C26 in GL4(𝔽53) generated by

 1 0 0 0 0 52 0 0 0 0 1 0 0 0 0 1
,
 52 0 0 0 0 52 0 0 0 0 6 0 0 0 0 6
,
 1 0 0 0 0 52 0 0 0 0 39 2 0 0 34 14
,
 1 0 0 0 0 52 0 0 0 0 1 0 0 0 14 52
G:=sub<GL(4,GF(53))| [1,0,0,0,0,52,0,0,0,0,1,0,0,0,0,1],[52,0,0,0,0,52,0,0,0,0,6,0,0,0,0,6],[1,0,0,0,0,52,0,0,0,0,39,34,0,0,2,14],[1,0,0,0,0,52,0,0,0,0,1,14,0,0,0,52] >;

D4×C2×C26 in GAP, Magma, Sage, TeX

D_4\times C_2\times C_{26}
% in TeX

G:=Group("D4xC2xC26");
// GroupNames label

G:=SmallGroup(416,228);
// by ID

G=gap.SmallGroup(416,228);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-13,-2,2521]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^26=c^4=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

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