direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C2×C26, C24⋊2C26, C52⋊4C23, C26.16C24, C4⋊(C22×C26), (C2×C26)⋊2C23, (C23×C26)⋊2C2, (C22×C4)⋊5C26, C23⋊3(C2×C26), C22⋊(C22×C26), (C22×C52)⋊12C2, (C2×C52)⋊15C22, C2.1(C23×C26), (C22×C26)⋊6C22, (C2×C4)⋊4(C2×C26), SmallGroup(416,228)
Series: Derived ►Chief ►Lower central ►Upper central
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, C2, C4, C22, C22, C2×C4, D4, C23, C23, C23, C13, C22×C4, C2×D4, C24, C26, C26, C26, C22×D4, C52, C2×C26, C2×C26, C2×C52, D4×C13, C22×C26, C22×C26, C22×C26, C22×C52, D4×C26, C23×C26, D4×C2×C26
Quotients: C1, C2, C22, D4, C23, C13, C2×D4, C24, C26, C22×D4, C2×C26, D4×C13, C22×C26, D4×C26, C23×C26, D4×C2×C26
►On 208 points
Generators in S208
(1 138)(2 139)(3 140)(4 141)(5 142)(6 143)(7 144)(8 145)(9 146)(10 147)(11 148)(12 149)(13 150)(14 151)(15 152)(16 153)(17 154)(18 155)(19 156)(20 131)(21 132)(22 133)(23 134)(24 135)(25 136)(26 137)(27 174)(28 175)(29 176)(30 177)(31 178)(32 179)(33 180)(34 181)(35 182)(36 157)(37 158)(38 159)(39 160)(40 161)(41 162)(42 163)(43 164)(44 165)(45 166)(46 167)(47 168)(48 169)(49 170)(50 171)(51 172)(52 173)(53 188)(54 189)(55 190)(56 191)(57 192)(58 193)(59 194)(60 195)(61 196)(62 197)(63 198)(64 199)(65 200)(66 201)(67 202)(68 203)(69 204)(70 205)(71 206)(72 207)(73 208)(74 183)(75 184)(76 185)(77 186)(78 187)(79 127)(80 128)(81 129)(82 130)(83 105)(84 106)(85 107)(86 108)(87 109)(88 110)(89 111)(90 112)(91 113)(92 114)(93 115)(94 116)(95 117)(96 118)(97 119)(98 120)(99 121)(100 122)(101 123)(102 124)(103 125)(104 126)
(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)
(1 200 158 105)(2 201 159 106)(3 202 160 107)(4 203 161 108)(5 204 162 109)(6 205 163 110)(7 206 164 111)(8 207 165 112)(9 208 166 113)(10 183 167 114)(11 184 168 115)(12 185 169 116)(13 186 170 117)(14 187 171 118)(15 188 172 119)(16 189 173 120)(17 190 174 121)(18 191 175 122)(19 192 176 123)(20 193 177 124)(21 194 178 125)(22 195 179 126)(23 196 180 127)(24 197 181 128)(25 198 182 129)(26 199 157 130)(27 99 154 55)(28 100 155 56)(29 101 156 57)(30 102 131 58)(31 103 132 59)(32 104 133 60)(33 79 134 61)(34 80 135 62)(35 81 136 63)(36 82 137 64)(37 83 138 65)(38 84 139 66)(39 85 140 67)(40 86 141 68)(41 87 142 69)(42 88 143 70)(43 89 144 71)(44 90 145 72)(45 91 146 73)(46 92 147 74)(47 93 148 75)(48 94 149 76)(49 95 150 77)(50 96 151 78)(51 97 152 53)(52 98 153 54)
(1 50)(2 51)(3 52)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 40)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 48)(26 49)(53 201)(54 202)(55 203)(56 204)(57 205)(58 206)(59 207)(60 208)(61 183)(62 184)(63 185)(64 186)(65 187)(66 188)(67 189)(68 190)(69 191)(70 192)(71 193)(72 194)(73 195)(74 196)(75 197)(76 198)(77 199)(78 200)(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)(96 105)(97 106)(98 107)(99 108)(100 109)(101 110)(102 111)(103 112)(104 113)(131 164)(132 165)(133 166)(134 167)(135 168)(136 169)(137 170)(138 171)(139 172)(140 173)(141 174)(142 175)(143 176)(144 177)(145 178)(146 179)(147 180)(148 181)(149 182)(150 157)(151 158)(152 159)(153 160)(154 161)(155 162)(156 163)
G:=sub<Sym(208)| (1,138)(2,139)(3,140)(4,141)(5,142)(6,143)(7,144)(8,145)(9,146)(10,147)(11,148)(12,149)(13,150)(14,151)(15,152)(16,153)(17,154)(18,155)(19,156)(20,131)(21,132)(22,133)(23,134)(24,135)(25,136)(26,137)(27,174)(28,175)(29,176)(30,177)(31,178)(32,179)(33,180)(34,181)(35,182)(36,157)(37,158)(38,159)(39,160)(40,161)(41,162)(42,163)(43,164)(44,165)(45,166)(46,167)(47,168)(48,169)(49,170)(50,171)(51,172)(52,173)(53,188)(54,189)(55,190)(56,191)(57,192)(58,193)(59,194)(60,195)(61,196)(62,197)(63,198)(64,199)(65,200)(66,201)(67,202)(68,203)(69,204)(70,205)(71,206)(72,207)(73,208)(74,183)(75,184)(76,185)(77,186)(78,187)(79,127)(80,128)(81,129)(82,130)(83,105)(84,106)(85,107)(86,108)(87,109)(88,110)(89,111)(90,112)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(97,119)(98,120)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126), (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), (1,200,158,105)(2,201,159,106)(3,202,160,107)(4,203,161,108)(5,204,162,109)(6,205,163,110)(7,206,164,111)(8,207,165,112)(9,208,166,113)(10,183,167,114)(11,184,168,115)(12,185,169,116)(13,186,170,117)(14,187,171,118)(15,188,172,119)(16,189,173,120)(17,190,174,121)(18,191,175,122)(19,192,176,123)(20,193,177,124)(21,194,178,125)(22,195,179,126)(23,196,180,127)(24,197,181,128)(25,198,182,129)(26,199,157,130)(27,99,154,55)(28,100,155,56)(29,101,156,57)(30,102,131,58)(31,103,132,59)(32,104,133,60)(33,79,134,61)(34,80,135,62)(35,81,136,63)(36,82,137,64)(37,83,138,65)(38,84,139,66)(39,85,140,67)(40,86,141,68)(41,87,142,69)(42,88,143,70)(43,89,144,71)(44,90,145,72)(45,91,146,73)(46,92,147,74)(47,93,148,75)(48,94,149,76)(49,95,150,77)(50,96,151,78)(51,97,152,53)(52,98,153,54), (1,50)(2,51)(3,52)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(53,201)(54,202)(55,203)(56,204)(57,205)(58,206)(59,207)(60,208)(61,183)(62,184)(63,185)(64,186)(65,187)(66,188)(67,189)(68,190)(69,191)(70,192)(71,193)(72,194)(73,195)(74,196)(75,197)(76,198)(77,199)(78,200)(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)(96,105)(97,106)(98,107)(99,108)(100,109)(101,110)(102,111)(103,112)(104,113)(131,164)(132,165)(133,166)(134,167)(135,168)(136,169)(137,170)(138,171)(139,172)(140,173)(141,174)(142,175)(143,176)(144,177)(145,178)(146,179)(147,180)(148,181)(149,182)(150,157)(151,158)(152,159)(153,160)(154,161)(155,162)(156,163)>;
G:=Group( (1,138)(2,139)(3,140)(4,141)(5,142)(6,143)(7,144)(8,145)(9,146)(10,147)(11,148)(12,149)(13,150)(14,151)(15,152)(16,153)(17,154)(18,155)(19,156)(20,131)(21,132)(22,133)(23,134)(24,135)(25,136)(26,137)(27,174)(28,175)(29,176)(30,177)(31,178)(32,179)(33,180)(34,181)(35,182)(36,157)(37,158)(38,159)(39,160)(40,161)(41,162)(42,163)(43,164)(44,165)(45,166)(46,167)(47,168)(48,169)(49,170)(50,171)(51,172)(52,173)(53,188)(54,189)(55,190)(56,191)(57,192)(58,193)(59,194)(60,195)(61,196)(62,197)(63,198)(64,199)(65,200)(66,201)(67,202)(68,203)(69,204)(70,205)(71,206)(72,207)(73,208)(74,183)(75,184)(76,185)(77,186)(78,187)(79,127)(80,128)(81,129)(82,130)(83,105)(84,106)(85,107)(86,108)(87,109)(88,110)(89,111)(90,112)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(97,119)(98,120)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126), (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), (1,200,158,105)(2,201,159,106)(3,202,160,107)(4,203,161,108)(5,204,162,109)(6,205,163,110)(7,206,164,111)(8,207,165,112)(9,208,166,113)(10,183,167,114)(11,184,168,115)(12,185,169,116)(13,186,170,117)(14,187,171,118)(15,188,172,119)(16,189,173,120)(17,190,174,121)(18,191,175,122)(19,192,176,123)(20,193,177,124)(21,194,178,125)(22,195,179,126)(23,196,180,127)(24,197,181,128)(25,198,182,129)(26,199,157,130)(27,99,154,55)(28,100,155,56)(29,101,156,57)(30,102,131,58)(31,103,132,59)(32,104,133,60)(33,79,134,61)(34,80,135,62)(35,81,136,63)(36,82,137,64)(37,83,138,65)(38,84,139,66)(39,85,140,67)(40,86,141,68)(41,87,142,69)(42,88,143,70)(43,89,144,71)(44,90,145,72)(45,91,146,73)(46,92,147,74)(47,93,148,75)(48,94,149,76)(49,95,150,77)(50,96,151,78)(51,97,152,53)(52,98,153,54), (1,50)(2,51)(3,52)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(53,201)(54,202)(55,203)(56,204)(57,205)(58,206)(59,207)(60,208)(61,183)(62,184)(63,185)(64,186)(65,187)(66,188)(67,189)(68,190)(69,191)(70,192)(71,193)(72,194)(73,195)(74,196)(75,197)(76,198)(77,199)(78,200)(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)(96,105)(97,106)(98,107)(99,108)(100,109)(101,110)(102,111)(103,112)(104,113)(131,164)(132,165)(133,166)(134,167)(135,168)(136,169)(137,170)(138,171)(139,172)(140,173)(141,174)(142,175)(143,176)(144,177)(145,178)(146,179)(147,180)(148,181)(149,182)(150,157)(151,158)(152,159)(153,160)(154,161)(155,162)(156,163) );
G=PermutationGroup([[(1,138),(2,139),(3,140),(4,141),(5,142),(6,143),(7,144),(8,145),(9,146),(10,147),(11,148),(12,149),(13,150),(14,151),(15,152),(16,153),(17,154),(18,155),(19,156),(20,131),(21,132),(22,133),(23,134),(24,135),(25,136),(26,137),(27,174),(28,175),(29,176),(30,177),(31,178),(32,179),(33,180),(34,181),(35,182),(36,157),(37,158),(38,159),(39,160),(40,161),(41,162),(42,163),(43,164),(44,165),(45,166),(46,167),(47,168),(48,169),(49,170),(50,171),(51,172),(52,173),(53,188),(54,189),(55,190),(56,191),(57,192),(58,193),(59,194),(60,195),(61,196),(62,197),(63,198),(64,199),(65,200),(66,201),(67,202),(68,203),(69,204),(70,205),(71,206),(72,207),(73,208),(74,183),(75,184),(76,185),(77,186),(78,187),(79,127),(80,128),(81,129),(82,130),(83,105),(84,106),(85,107),(86,108),(87,109),(88,110),(89,111),(90,112),(91,113),(92,114),(93,115),(94,116),(95,117),(96,118),(97,119),(98,120),(99,121),(100,122),(101,123),(102,124),(103,125),(104,126)], [(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)], [(1,200,158,105),(2,201,159,106),(3,202,160,107),(4,203,161,108),(5,204,162,109),(6,205,163,110),(7,206,164,111),(8,207,165,112),(9,208,166,113),(10,183,167,114),(11,184,168,115),(12,185,169,116),(13,186,170,117),(14,187,171,118),(15,188,172,119),(16,189,173,120),(17,190,174,121),(18,191,175,122),(19,192,176,123),(20,193,177,124),(21,194,178,125),(22,195,179,126),(23,196,180,127),(24,197,181,128),(25,198,182,129),(26,199,157,130),(27,99,154,55),(28,100,155,56),(29,101,156,57),(30,102,131,58),(31,103,132,59),(32,104,133,60),(33,79,134,61),(34,80,135,62),(35,81,136,63),(36,82,137,64),(37,83,138,65),(38,84,139,66),(39,85,140,67),(40,86,141,68),(41,87,142,69),(42,88,143,70),(43,89,144,71),(44,90,145,72),(45,91,146,73),(46,92,147,74),(47,93,148,75),(48,94,149,76),(49,95,150,77),(50,96,151,78),(51,97,152,53),(52,98,153,54)], [(1,50),(2,51),(3,52),(4,27),(5,28),(6,29),(7,30),(8,31),(9,32),(10,33),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(17,40),(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47),(25,48),(26,49),(53,201),(54,202),(55,203),(56,204),(57,205),(58,206),(59,207),(60,208),(61,183),(62,184),(63,185),(64,186),(65,187),(66,188),(67,189),(68,190),(69,191),(70,192),(71,193),(72,194),(73,195),(74,196),(75,197),(76,198),(77,199),(78,200),(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),(96,105),(97,106),(98,107),(99,108),(100,109),(101,110),(102,111),(103,112),(104,113),(131,164),(132,165),(133,166),(134,167),(135,168),(136,169),(137,170),(138,171),(139,172),(140,173),(141,174),(142,175),(143,176),(144,177),(145,178),(146,179),(147,180),(148,181),(149,182),(150,157),(151,158),(152,159),(153,160),(154,161),(155,162),(156,163)]])
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