metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D26.12D4, C23.4D26, C52⋊3C4⋊4C2, C2.8(D4×D13), (C2×C4).6D26, C22⋊C4⋊3D13, C26.19(C2×D4), D26⋊C4⋊5C2, C26.8(C4○D4), C23.D13⋊4C2, C26.D4⋊10C2, (C2×C26).24C23, (C2×C52).52C22, C2.8(D4⋊2D13), C13⋊1(C22.D4), C2.10(D52⋊5C2), (C22×C26).13C22, (C2×Dic13).6C22, C22.42(C22×D13), (C22×D13).20C22, (C2×C4×D13)⋊10C2, (C13×C22⋊C4)⋊5C2, (C2×C13⋊D4).3C2, SmallGroup(416,104)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D26.12D4
G = < a,b,c,d | a26=b2=c4=1, d2=a13, bab=a-1, ac=ca, ad=da, cbc-1=a13b, bd=db, dcd-1=a13c-1 >
Subgroups: 536 in 78 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C13, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D13, C26, C26, C22.D4, Dic13, C52, D26, D26, C2×C26, C2×C26, C4×D13, C2×Dic13, C13⋊D4, C2×C52, C22×D13, C22×C26, C26.D4, C52⋊3C4, D26⋊C4, C23.D13, C13×C22⋊C4, C2×C4×D13, C2×C13⋊D4, D26.12D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D13, C22.D4, D26, C22×D13, D52⋊5C2, D4×D13, D4⋊2D13, D26.12D4
►On 208 points
Generators in S208
(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 27)(2 52)(3 51)(4 50)(5 49)(6 48)(7 47)(8 46)(9 45)(10 44)(11 43)(12 42)(13 41)(14 40)(15 39)(16 38)(17 37)(18 36)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(53 87)(54 86)(55 85)(56 84)(57 83)(58 82)(59 81)(60 80)(61 79)(62 104)(63 103)(64 102)(65 101)(66 100)(67 99)(68 98)(69 97)(70 96)(71 95)(72 94)(73 93)(74 92)(75 91)(76 90)(77 89)(78 88)(105 194)(106 193)(107 192)(108 191)(109 190)(110 189)(111 188)(112 187)(113 186)(114 185)(115 184)(116 183)(117 208)(118 207)(119 206)(120 205)(121 204)(122 203)(123 202)(124 201)(125 200)(126 199)(127 198)(128 197)(129 196)(130 195)(131 172)(132 171)(133 170)(134 169)(135 168)(136 167)(137 166)(138 165)(139 164)(140 163)(141 162)(142 161)(143 160)(144 159)(145 158)(146 157)(147 182)(148 181)(149 180)(150 179)(151 178)(152 177)(153 176)(154 175)(155 174)(156 173)
(1 135 41 169)(2 136 42 170)(3 137 43 171)(4 138 44 172)(5 139 45 173)(6 140 46 174)(7 141 47 175)(8 142 48 176)(9 143 49 177)(10 144 50 178)(11 145 51 179)(12 146 52 180)(13 147 27 181)(14 148 28 182)(15 149 29 157)(16 150 30 158)(17 151 31 159)(18 152 32 160)(19 153 33 161)(20 154 34 162)(21 155 35 163)(22 156 36 164)(23 131 37 165)(24 132 38 166)(25 133 39 167)(26 134 40 168)(53 192 81 114)(54 193 82 115)(55 194 83 116)(56 195 84 117)(57 196 85 118)(58 197 86 119)(59 198 87 120)(60 199 88 121)(61 200 89 122)(62 201 90 123)(63 202 91 124)(64 203 92 125)(65 204 93 126)(66 205 94 127)(67 206 95 128)(68 207 96 129)(69 208 97 130)(70 183 98 105)(71 184 99 106)(72 185 100 107)(73 186 101 108)(74 187 102 109)(75 188 103 110)(76 189 104 111)(77 190 79 112)(78 191 80 113)
(1 76 14 63)(2 77 15 64)(3 78 16 65)(4 53 17 66)(5 54 18 67)(6 55 19 68)(7 56 20 69)(8 57 21 70)(9 58 22 71)(10 59 23 72)(11 60 24 73)(12 61 25 74)(13 62 26 75)(27 90 40 103)(28 91 41 104)(29 92 42 79)(30 93 43 80)(31 94 44 81)(32 95 45 82)(33 96 46 83)(34 97 47 84)(35 98 48 85)(36 99 49 86)(37 100 50 87)(38 101 51 88)(39 102 52 89)(105 155 118 142)(106 156 119 143)(107 131 120 144)(108 132 121 145)(109 133 122 146)(110 134 123 147)(111 135 124 148)(112 136 125 149)(113 137 126 150)(114 138 127 151)(115 139 128 152)(116 140 129 153)(117 141 130 154)(157 190 170 203)(158 191 171 204)(159 192 172 205)(160 193 173 206)(161 194 174 207)(162 195 175 208)(163 196 176 183)(164 197 177 184)(165 198 178 185)(166 199 179 186)(167 200 180 187)(168 201 181 188)(169 202 182 189)
G:=sub<Sym(208)| (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), (1,27)(2,52)(3,51)(4,50)(5,49)(6,48)(7,47)(8,46)(9,45)(10,44)(11,43)(12,42)(13,41)(14,40)(15,39)(16,38)(17,37)(18,36)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(53,87)(54,86)(55,85)(56,84)(57,83)(58,82)(59,81)(60,80)(61,79)(62,104)(63,103)(64,102)(65,101)(66,100)(67,99)(68,98)(69,97)(70,96)(71,95)(72,94)(73,93)(74,92)(75,91)(76,90)(77,89)(78,88)(105,194)(106,193)(107,192)(108,191)(109,190)(110,189)(111,188)(112,187)(113,186)(114,185)(115,184)(116,183)(117,208)(118,207)(119,206)(120,205)(121,204)(122,203)(123,202)(124,201)(125,200)(126,199)(127,198)(128,197)(129,196)(130,195)(131,172)(132,171)(133,170)(134,169)(135,168)(136,167)(137,166)(138,165)(139,164)(140,163)(141,162)(142,161)(143,160)(144,159)(145,158)(146,157)(147,182)(148,181)(149,180)(150,179)(151,178)(152,177)(153,176)(154,175)(155,174)(156,173), (1,135,41,169)(2,136,42,170)(3,137,43,171)(4,138,44,172)(5,139,45,173)(6,140,46,174)(7,141,47,175)(8,142,48,176)(9,143,49,177)(10,144,50,178)(11,145,51,179)(12,146,52,180)(13,147,27,181)(14,148,28,182)(15,149,29,157)(16,150,30,158)(17,151,31,159)(18,152,32,160)(19,153,33,161)(20,154,34,162)(21,155,35,163)(22,156,36,164)(23,131,37,165)(24,132,38,166)(25,133,39,167)(26,134,40,168)(53,192,81,114)(54,193,82,115)(55,194,83,116)(56,195,84,117)(57,196,85,118)(58,197,86,119)(59,198,87,120)(60,199,88,121)(61,200,89,122)(62,201,90,123)(63,202,91,124)(64,203,92,125)(65,204,93,126)(66,205,94,127)(67,206,95,128)(68,207,96,129)(69,208,97,130)(70,183,98,105)(71,184,99,106)(72,185,100,107)(73,186,101,108)(74,187,102,109)(75,188,103,110)(76,189,104,111)(77,190,79,112)(78,191,80,113), (1,76,14,63)(2,77,15,64)(3,78,16,65)(4,53,17,66)(5,54,18,67)(6,55,19,68)(7,56,20,69)(8,57,21,70)(9,58,22,71)(10,59,23,72)(11,60,24,73)(12,61,25,74)(13,62,26,75)(27,90,40,103)(28,91,41,104)(29,92,42,79)(30,93,43,80)(31,94,44,81)(32,95,45,82)(33,96,46,83)(34,97,47,84)(35,98,48,85)(36,99,49,86)(37,100,50,87)(38,101,51,88)(39,102,52,89)(105,155,118,142)(106,156,119,143)(107,131,120,144)(108,132,121,145)(109,133,122,146)(110,134,123,147)(111,135,124,148)(112,136,125,149)(113,137,126,150)(114,138,127,151)(115,139,128,152)(116,140,129,153)(117,141,130,154)(157,190,170,203)(158,191,171,204)(159,192,172,205)(160,193,173,206)(161,194,174,207)(162,195,175,208)(163,196,176,183)(164,197,177,184)(165,198,178,185)(166,199,179,186)(167,200,180,187)(168,201,181,188)(169,202,182,189)>;
G:=Group( (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), (1,27)(2,52)(3,51)(4,50)(5,49)(6,48)(7,47)(8,46)(9,45)(10,44)(11,43)(12,42)(13,41)(14,40)(15,39)(16,38)(17,37)(18,36)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(53,87)(54,86)(55,85)(56,84)(57,83)(58,82)(59,81)(60,80)(61,79)(62,104)(63,103)(64,102)(65,101)(66,100)(67,99)(68,98)(69,97)(70,96)(71,95)(72,94)(73,93)(74,92)(75,91)(76,90)(77,89)(78,88)(105,194)(106,193)(107,192)(108,191)(109,190)(110,189)(111,188)(112,187)(113,186)(114,185)(115,184)(116,183)(117,208)(118,207)(119,206)(120,205)(121,204)(122,203)(123,202)(124,201)(125,200)(126,199)(127,198)(128,197)(129,196)(130,195)(131,172)(132,171)(133,170)(134,169)(135,168)(136,167)(137,166)(138,165)(139,164)(140,163)(141,162)(142,161)(143,160)(144,159)(145,158)(146,157)(147,182)(148,181)(149,180)(150,179)(151,178)(152,177)(153,176)(154,175)(155,174)(156,173), (1,135,41,169)(2,136,42,170)(3,137,43,171)(4,138,44,172)(5,139,45,173)(6,140,46,174)(7,141,47,175)(8,142,48,176)(9,143,49,177)(10,144,50,178)(11,145,51,179)(12,146,52,180)(13,147,27,181)(14,148,28,182)(15,149,29,157)(16,150,30,158)(17,151,31,159)(18,152,32,160)(19,153,33,161)(20,154,34,162)(21,155,35,163)(22,156,36,164)(23,131,37,165)(24,132,38,166)(25,133,39,167)(26,134,40,168)(53,192,81,114)(54,193,82,115)(55,194,83,116)(56,195,84,117)(57,196,85,118)(58,197,86,119)(59,198,87,120)(60,199,88,121)(61,200,89,122)(62,201,90,123)(63,202,91,124)(64,203,92,125)(65,204,93,126)(66,205,94,127)(67,206,95,128)(68,207,96,129)(69,208,97,130)(70,183,98,105)(71,184,99,106)(72,185,100,107)(73,186,101,108)(74,187,102,109)(75,188,103,110)(76,189,104,111)(77,190,79,112)(78,191,80,113), (1,76,14,63)(2,77,15,64)(3,78,16,65)(4,53,17,66)(5,54,18,67)(6,55,19,68)(7,56,20,69)(8,57,21,70)(9,58,22,71)(10,59,23,72)(11,60,24,73)(12,61,25,74)(13,62,26,75)(27,90,40,103)(28,91,41,104)(29,92,42,79)(30,93,43,80)(31,94,44,81)(32,95,45,82)(33,96,46,83)(34,97,47,84)(35,98,48,85)(36,99,49,86)(37,100,50,87)(38,101,51,88)(39,102,52,89)(105,155,118,142)(106,156,119,143)(107,131,120,144)(108,132,121,145)(109,133,122,146)(110,134,123,147)(111,135,124,148)(112,136,125,149)(113,137,126,150)(114,138,127,151)(115,139,128,152)(116,140,129,153)(117,141,130,154)(157,190,170,203)(158,191,171,204)(159,192,172,205)(160,193,173,206)(161,194,174,207)(162,195,175,208)(163,196,176,183)(164,197,177,184)(165,198,178,185)(166,199,179,186)(167,200,180,187)(168,201,181,188)(169,202,182,189) );
G=PermutationGroup([[(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)], [(1,27),(2,52),(3,51),(4,50),(5,49),(6,48),(7,47),(8,46),(9,45),(10,44),(11,43),(12,42),(13,41),(14,40),(15,39),(16,38),(17,37),(18,36),(19,35),(20,34),(21,33),(22,32),(23,31),(24,30),(25,29),(26,28),(53,87),(54,86),(55,85),(56,84),(57,83),(58,82),(59,81),(60,80),(61,79),(62,104),(63,103),(64,102),(65,101),(66,100),(67,99),(68,98),(69,97),(70,96),(71,95),(72,94),(73,93),(74,92),(75,91),(76,90),(77,89),(78,88),(105,194),(106,193),(107,192),(108,191),(109,190),(110,189),(111,188),(112,187),(113,186),(114,185),(115,184),(116,183),(117,208),(118,207),(119,206),(120,205),(121,204),(122,203),(123,202),(124,201),(125,200),(126,199),(127,198),(128,197),(129,196),(130,195),(131,172),(132,171),(133,170),(134,169),(135,168),(136,167),(137,166),(138,165),(139,164),(140,163),(141,162),(142,161),(143,160),(144,159),(145,158),(146,157),(147,182),(148,181),(149,180),(150,179),(151,178),(152,177),(153,176),(154,175),(155,174),(156,173)], [(1,135,41,169),(2,136,42,170),(3,137,43,171),(4,138,44,172),(5,139,45,173),(6,140,46,174),(7,141,47,175),(8,142,48,176),(9,143,49,177),(10,144,50,178),(11,145,51,179),(12,146,52,180),(13,147,27,181),(14,148,28,182),(15,149,29,157),(16,150,30,158),(17,151,31,159),(18,152,32,160),(19,153,33,161),(20,154,34,162),(21,155,35,163),(22,156,36,164),(23,131,37,165),(24,132,38,166),(25,133,39,167),(26,134,40,168),(53,192,81,114),(54,193,82,115),(55,194,83,116),(56,195,84,117),(57,196,85,118),(58,197,86,119),(59,198,87,120),(60,199,88,121),(61,200,89,122),(62,201,90,123),(63,202,91,124),(64,203,92,125),(65,204,93,126),(66,205,94,127),(67,206,95,128),(68,207,96,129),(69,208,97,130),(70,183,98,105),(71,184,99,106),(72,185,100,107),(73,186,101,108),(74,187,102,109),(75,188,103,110),(76,189,104,111),(77,190,79,112),(78,191,80,113)], [(1,76,14,63),(2,77,15,64),(3,78,16,65),(4,53,17,66),(5,54,18,67),(6,55,19,68),(7,56,20,69),(8,57,21,70),(9,58,22,71),(10,59,23,72),(11,60,24,73),(12,61,25,74),(13,62,26,75),(27,90,40,103),(28,91,41,104),(29,92,42,79),(30,93,43,80),(31,94,44,81),(32,95,45,82),(33,96,46,83),(34,97,47,84),(35,98,48,85),(36,99,49,86),(37,100,50,87),(38,101,51,88),(39,102,52,89),(105,155,118,142),(106,156,119,143),(107,131,120,144),(108,132,121,145),(109,133,122,146),(110,134,123,147),(111,135,124,148),(112,136,125,149),(113,137,126,150),(114,138,127,151),(115,139,128,152),(116,140,129,153),(117,141,130,154),(157,190,170,203),(158,191,171,204),(159,192,172,205),(160,193,173,206),(161,194,174,207),(162,195,175,208),(163,196,176,183),(164,197,177,184),(165,198,178,185),(166,199,179,186),(167,200,180,187),(168,201,181,188),(169,202,182,189)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 13A | ··· | 13F | 26A | ··· | 26R | 26S | ··· | 26AD | 52A | ··· | 52X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 4 | 26 | 26 | 2 | 2 | 4 | 26 | 26 | 52 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | D13 | D26 | D26 | D52⋊5C2 | D4×D13 | D4⋊2D13 |
| kernel | D26.12D4 | C26.D4 | C52⋊3C4 | D26⋊C4 | C23.D13 | C13×C22⋊C4 | C2×C4×D13 | C2×C13⋊D4 | D26 | C26 | C22⋊C4 | C2×C4 | C23 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 6 | 12 | 6 | 24 | 6 | 6 |
Matrix representation of D26.12D4 ►in GL4(𝔽53) generated by
| 16 | 46 | 0 | 0 |
| 10 | 52 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
| 26 | 31 | 0 | 0 |
| 9 | 27 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 27 | 52 |
| 42 | 34 | 0 | 0 |
| 12 | 11 | 0 | 0 |
| 0 | 0 | 23 | 14 |
| 0 | 0 | 38 | 30 |
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 23 | 0 |
| 0 | 0 | 38 | 30 |
G:=sub<GL(4,GF(53))| [16,10,0,0,46,52,0,0,0,0,52,0,0,0,0,52],[26,9,0,0,31,27,0,0,0,0,1,27,0,0,0,52],[42,12,0,0,34,11,0,0,0,0,23,38,0,0,14,30],[30,0,0,0,0,30,0,0,0,0,23,38,0,0,0,30] >;
D26.12D4 in GAP, Magma, Sage, TeX
D_{26}._{12}D_4 % in TeX
G:=Group("D26.12D4"); // GroupNames label
G:=SmallGroup(416,104);
// by ID
G=gap.SmallGroup(416,104);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,55,218,188,13829]);
// Polycyclic
G:=Group<a,b,c,d|a^26=b^2=c^4=1,d^2=a^13,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^13*b,b*d=d*b,d*c*d^-1=a^13*c^-1>;
// generators/relations