metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D26⋊3Q8, C52.22D4, (Q8×C26)⋊3C2, (C2×Q8)⋊3D13, C2.9(Q8×D13), C52⋊3C4⋊15C2, (C2×C4).21D26, C26.57(C2×D4), C13⋊5(C22⋊Q8), C26.17(C2×Q8), D26⋊C4.6C2, C26.36(C4○D4), C26.D4⋊16C2, C4.18(C13⋊D4), (C2×C26).58C23, (C2×C52).64C22, C2.8(D52⋊C2), C22.64(C22×D13), (C2×Dic13).21C22, (C22×D13).30C22, (C2×C4×D13).5C2, C2.21(C2×C13⋊D4), SmallGroup(416,167)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D26⋊3Q8
G = < a,b,c,d | a26=b2=c4=1, d2=c2, bab=a-1, ac=ca, ad=da, cbc-1=a13b, bd=db, dcd-1=c-1 >
Subgroups: 472 in 74 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, Q8, C23, C13, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, D13, C26, C22⋊Q8, Dic13, C52, C52, D26, D26, C2×C26, C4×D13, C2×Dic13, C2×Dic13, C2×C52, C2×C52, Q8×C13, C22×D13, C26.D4, C52⋊3C4, D26⋊C4, C2×C4×D13, Q8×C26, D26⋊3Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, D13, C22⋊Q8, D26, C13⋊D4, C22×D13, Q8×D13, D52⋊C2, C2×C13⋊D4, D26⋊3Q8
►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)
(1 154)(2 153)(3 152)(4 151)(5 150)(6 149)(7 148)(8 147)(9 146)(10 145)(11 144)(12 143)(13 142)(14 141)(15 140)(16 139)(17 138)(18 137)(19 136)(20 135)(21 134)(22 133)(23 132)(24 131)(25 156)(26 155)(27 169)(28 168)(29 167)(30 166)(31 165)(32 164)(33 163)(34 162)(35 161)(36 160)(37 159)(38 158)(39 157)(40 182)(41 181)(42 180)(43 179)(44 178)(45 177)(46 176)(47 175)(48 174)(49 173)(50 172)(51 171)(52 170)(53 109)(54 108)(55 107)(56 106)(57 105)(58 130)(59 129)(60 128)(61 127)(62 126)(63 125)(64 124)(65 123)(66 122)(67 121)(68 120)(69 119)(70 118)(71 117)(72 116)(73 115)(74 114)(75 113)(76 112)(77 111)(78 110)(79 205)(80 204)(81 203)(82 202)(83 201)(84 200)(85 199)(86 198)(87 197)(88 196)(89 195)(90 194)(91 193)(92 192)(93 191)(94 190)(95 189)(96 188)(97 187)(98 186)(99 185)(100 184)(101 183)(102 208)(103 207)(104 206)
(1 183 155 89)(2 184 156 90)(3 185 131 91)(4 186 132 92)(5 187 133 93)(6 188 134 94)(7 189 135 95)(8 190 136 96)(9 191 137 97)(10 192 138 98)(11 193 139 99)(12 194 140 100)(13 195 141 101)(14 196 142 102)(15 197 143 103)(16 198 144 104)(17 199 145 79)(18 200 146 80)(19 201 147 81)(20 202 148 82)(21 203 149 83)(22 204 150 84)(23 205 151 85)(24 206 152 86)(25 207 153 87)(26 208 154 88)(27 67 177 116)(28 68 178 117)(29 69 179 118)(30 70 180 119)(31 71 181 120)(32 72 182 121)(33 73 157 122)(34 74 158 123)(35 75 159 124)(36 76 160 125)(37 77 161 126)(38 78 162 127)(39 53 163 128)(40 54 164 129)(41 55 165 130)(42 56 166 105)(43 57 167 106)(44 58 168 107)(45 59 169 108)(46 60 170 109)(47 61 171 110)(48 62 172 111)(49 63 173 112)(50 64 174 113)(51 65 175 114)(52 66 176 115)
(1 57 155 106)(2 58 156 107)(3 59 131 108)(4 60 132 109)(5 61 133 110)(6 62 134 111)(7 63 135 112)(8 64 136 113)(9 65 137 114)(10 66 138 115)(11 67 139 116)(12 68 140 117)(13 69 141 118)(14 70 142 119)(15 71 143 120)(16 72 144 121)(17 73 145 122)(18 74 146 123)(19 75 147 124)(20 76 148 125)(21 77 149 126)(22 78 150 127)(23 53 151 128)(24 54 152 129)(25 55 153 130)(26 56 154 105)(27 99 177 193)(28 100 178 194)(29 101 179 195)(30 102 180 196)(31 103 181 197)(32 104 182 198)(33 79 157 199)(34 80 158 200)(35 81 159 201)(36 82 160 202)(37 83 161 203)(38 84 162 204)(39 85 163 205)(40 86 164 206)(41 87 165 207)(42 88 166 208)(43 89 167 183)(44 90 168 184)(45 91 169 185)(46 92 170 186)(47 93 171 187)(48 94 172 188)(49 95 173 189)(50 96 174 190)(51 97 175 191)(52 98 176 192)
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), (1,154)(2,153)(3,152)(4,151)(5,150)(6,149)(7,148)(8,147)(9,146)(10,145)(11,144)(12,143)(13,142)(14,141)(15,140)(16,139)(17,138)(18,137)(19,136)(20,135)(21,134)(22,133)(23,132)(24,131)(25,156)(26,155)(27,169)(28,168)(29,167)(30,166)(31,165)(32,164)(33,163)(34,162)(35,161)(36,160)(37,159)(38,158)(39,157)(40,182)(41,181)(42,180)(43,179)(44,178)(45,177)(46,176)(47,175)(48,174)(49,173)(50,172)(51,171)(52,170)(53,109)(54,108)(55,107)(56,106)(57,105)(58,130)(59,129)(60,128)(61,127)(62,126)(63,125)(64,124)(65,123)(66,122)(67,121)(68,120)(69,119)(70,118)(71,117)(72,116)(73,115)(74,114)(75,113)(76,112)(77,111)(78,110)(79,205)(80,204)(81,203)(82,202)(83,201)(84,200)(85,199)(86,198)(87,197)(88,196)(89,195)(90,194)(91,193)(92,192)(93,191)(94,190)(95,189)(96,188)(97,187)(98,186)(99,185)(100,184)(101,183)(102,208)(103,207)(104,206), (1,183,155,89)(2,184,156,90)(3,185,131,91)(4,186,132,92)(5,187,133,93)(6,188,134,94)(7,189,135,95)(8,190,136,96)(9,191,137,97)(10,192,138,98)(11,193,139,99)(12,194,140,100)(13,195,141,101)(14,196,142,102)(15,197,143,103)(16,198,144,104)(17,199,145,79)(18,200,146,80)(19,201,147,81)(20,202,148,82)(21,203,149,83)(22,204,150,84)(23,205,151,85)(24,206,152,86)(25,207,153,87)(26,208,154,88)(27,67,177,116)(28,68,178,117)(29,69,179,118)(30,70,180,119)(31,71,181,120)(32,72,182,121)(33,73,157,122)(34,74,158,123)(35,75,159,124)(36,76,160,125)(37,77,161,126)(38,78,162,127)(39,53,163,128)(40,54,164,129)(41,55,165,130)(42,56,166,105)(43,57,167,106)(44,58,168,107)(45,59,169,108)(46,60,170,109)(47,61,171,110)(48,62,172,111)(49,63,173,112)(50,64,174,113)(51,65,175,114)(52,66,176,115), (1,57,155,106)(2,58,156,107)(3,59,131,108)(4,60,132,109)(5,61,133,110)(6,62,134,111)(7,63,135,112)(8,64,136,113)(9,65,137,114)(10,66,138,115)(11,67,139,116)(12,68,140,117)(13,69,141,118)(14,70,142,119)(15,71,143,120)(16,72,144,121)(17,73,145,122)(18,74,146,123)(19,75,147,124)(20,76,148,125)(21,77,149,126)(22,78,150,127)(23,53,151,128)(24,54,152,129)(25,55,153,130)(26,56,154,105)(27,99,177,193)(28,100,178,194)(29,101,179,195)(30,102,180,196)(31,103,181,197)(32,104,182,198)(33,79,157,199)(34,80,158,200)(35,81,159,201)(36,82,160,202)(37,83,161,203)(38,84,162,204)(39,85,163,205)(40,86,164,206)(41,87,165,207)(42,88,166,208)(43,89,167,183)(44,90,168,184)(45,91,169,185)(46,92,170,186)(47,93,171,187)(48,94,172,188)(49,95,173,189)(50,96,174,190)(51,97,175,191)(52,98,176,192)>;
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), (1,154)(2,153)(3,152)(4,151)(5,150)(6,149)(7,148)(8,147)(9,146)(10,145)(11,144)(12,143)(13,142)(14,141)(15,140)(16,139)(17,138)(18,137)(19,136)(20,135)(21,134)(22,133)(23,132)(24,131)(25,156)(26,155)(27,169)(28,168)(29,167)(30,166)(31,165)(32,164)(33,163)(34,162)(35,161)(36,160)(37,159)(38,158)(39,157)(40,182)(41,181)(42,180)(43,179)(44,178)(45,177)(46,176)(47,175)(48,174)(49,173)(50,172)(51,171)(52,170)(53,109)(54,108)(55,107)(56,106)(57,105)(58,130)(59,129)(60,128)(61,127)(62,126)(63,125)(64,124)(65,123)(66,122)(67,121)(68,120)(69,119)(70,118)(71,117)(72,116)(73,115)(74,114)(75,113)(76,112)(77,111)(78,110)(79,205)(80,204)(81,203)(82,202)(83,201)(84,200)(85,199)(86,198)(87,197)(88,196)(89,195)(90,194)(91,193)(92,192)(93,191)(94,190)(95,189)(96,188)(97,187)(98,186)(99,185)(100,184)(101,183)(102,208)(103,207)(104,206), (1,183,155,89)(2,184,156,90)(3,185,131,91)(4,186,132,92)(5,187,133,93)(6,188,134,94)(7,189,135,95)(8,190,136,96)(9,191,137,97)(10,192,138,98)(11,193,139,99)(12,194,140,100)(13,195,141,101)(14,196,142,102)(15,197,143,103)(16,198,144,104)(17,199,145,79)(18,200,146,80)(19,201,147,81)(20,202,148,82)(21,203,149,83)(22,204,150,84)(23,205,151,85)(24,206,152,86)(25,207,153,87)(26,208,154,88)(27,67,177,116)(28,68,178,117)(29,69,179,118)(30,70,180,119)(31,71,181,120)(32,72,182,121)(33,73,157,122)(34,74,158,123)(35,75,159,124)(36,76,160,125)(37,77,161,126)(38,78,162,127)(39,53,163,128)(40,54,164,129)(41,55,165,130)(42,56,166,105)(43,57,167,106)(44,58,168,107)(45,59,169,108)(46,60,170,109)(47,61,171,110)(48,62,172,111)(49,63,173,112)(50,64,174,113)(51,65,175,114)(52,66,176,115), (1,57,155,106)(2,58,156,107)(3,59,131,108)(4,60,132,109)(5,61,133,110)(6,62,134,111)(7,63,135,112)(8,64,136,113)(9,65,137,114)(10,66,138,115)(11,67,139,116)(12,68,140,117)(13,69,141,118)(14,70,142,119)(15,71,143,120)(16,72,144,121)(17,73,145,122)(18,74,146,123)(19,75,147,124)(20,76,148,125)(21,77,149,126)(22,78,150,127)(23,53,151,128)(24,54,152,129)(25,55,153,130)(26,56,154,105)(27,99,177,193)(28,100,178,194)(29,101,179,195)(30,102,180,196)(31,103,181,197)(32,104,182,198)(33,79,157,199)(34,80,158,200)(35,81,159,201)(36,82,160,202)(37,83,161,203)(38,84,162,204)(39,85,163,205)(40,86,164,206)(41,87,165,207)(42,88,166,208)(43,89,167,183)(44,90,168,184)(45,91,169,185)(46,92,170,186)(47,93,171,187)(48,94,172,188)(49,95,173,189)(50,96,174,190)(51,97,175,191)(52,98,176,192) );
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)], [(1,154),(2,153),(3,152),(4,151),(5,150),(6,149),(7,148),(8,147),(9,146),(10,145),(11,144),(12,143),(13,142),(14,141),(15,140),(16,139),(17,138),(18,137),(19,136),(20,135),(21,134),(22,133),(23,132),(24,131),(25,156),(26,155),(27,169),(28,168),(29,167),(30,166),(31,165),(32,164),(33,163),(34,162),(35,161),(36,160),(37,159),(38,158),(39,157),(40,182),(41,181),(42,180),(43,179),(44,178),(45,177),(46,176),(47,175),(48,174),(49,173),(50,172),(51,171),(52,170),(53,109),(54,108),(55,107),(56,106),(57,105),(58,130),(59,129),(60,128),(61,127),(62,126),(63,125),(64,124),(65,123),(66,122),(67,121),(68,120),(69,119),(70,118),(71,117),(72,116),(73,115),(74,114),(75,113),(76,112),(77,111),(78,110),(79,205),(80,204),(81,203),(82,202),(83,201),(84,200),(85,199),(86,198),(87,197),(88,196),(89,195),(90,194),(91,193),(92,192),(93,191),(94,190),(95,189),(96,188),(97,187),(98,186),(99,185),(100,184),(101,183),(102,208),(103,207),(104,206)], [(1,183,155,89),(2,184,156,90),(3,185,131,91),(4,186,132,92),(5,187,133,93),(6,188,134,94),(7,189,135,95),(8,190,136,96),(9,191,137,97),(10,192,138,98),(11,193,139,99),(12,194,140,100),(13,195,141,101),(14,196,142,102),(15,197,143,103),(16,198,144,104),(17,199,145,79),(18,200,146,80),(19,201,147,81),(20,202,148,82),(21,203,149,83),(22,204,150,84),(23,205,151,85),(24,206,152,86),(25,207,153,87),(26,208,154,88),(27,67,177,116),(28,68,178,117),(29,69,179,118),(30,70,180,119),(31,71,181,120),(32,72,182,121),(33,73,157,122),(34,74,158,123),(35,75,159,124),(36,76,160,125),(37,77,161,126),(38,78,162,127),(39,53,163,128),(40,54,164,129),(41,55,165,130),(42,56,166,105),(43,57,167,106),(44,58,168,107),(45,59,169,108),(46,60,170,109),(47,61,171,110),(48,62,172,111),(49,63,173,112),(50,64,174,113),(51,65,175,114),(52,66,176,115)], [(1,57,155,106),(2,58,156,107),(3,59,131,108),(4,60,132,109),(5,61,133,110),(6,62,134,111),(7,63,135,112),(8,64,136,113),(9,65,137,114),(10,66,138,115),(11,67,139,116),(12,68,140,117),(13,69,141,118),(14,70,142,119),(15,71,143,120),(16,72,144,121),(17,73,145,122),(18,74,146,123),(19,75,147,124),(20,76,148,125),(21,77,149,126),(22,78,150,127),(23,53,151,128),(24,54,152,129),(25,55,153,130),(26,56,154,105),(27,99,177,193),(28,100,178,194),(29,101,179,195),(30,102,180,196),(31,103,181,197),(32,104,182,198),(33,79,157,199),(34,80,158,200),(35,81,159,201),(36,82,160,202),(37,83,161,203),(38,84,162,204),(39,85,163,205),(40,86,164,206),(41,87,165,207),(42,88,166,208),(43,89,167,183),(44,90,168,184),(45,91,169,185),(46,92,170,186),(47,93,171,187),(48,94,172,188),(49,95,173,189),(50,96,174,190),(51,97,175,191),(52,98,176,192)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 26 | 26 | 2 | 2 | 4 | 4 | 26 | 26 | 52 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | C4○D4 | D13 | D26 | C13⋊D4 | Q8×D13 | D52⋊C2 |
| kernel | D26⋊3Q8 | C26.D4 | C52⋊3C4 | D26⋊C4 | C2×C4×D13 | Q8×C26 | C52 | D26 | C26 | C2×Q8 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 6 | 18 | 24 | 6 | 6 |
Matrix representation of D26⋊3Q8 ►in GL4(𝔽53) generated by
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 33 | 4 |
| 0 | 0 | 31 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 20 | 52 |
| 0 | 1 | 0 | 0 |
| 52 | 0 | 0 | 0 |
| 0 | 0 | 25 | 24 |
| 0 | 0 | 27 | 28 |
| 30 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
G:=sub<GL(4,GF(53))| [52,0,0,0,0,52,0,0,0,0,33,31,0,0,4,7],[1,0,0,0,0,52,0,0,0,0,1,20,0,0,0,52],[0,52,0,0,1,0,0,0,0,0,25,27,0,0,24,28],[30,0,0,0,0,23,0,0,0,0,52,0,0,0,0,52] >;
D26⋊3Q8 in GAP, Magma, Sage, TeX
D_{26}\rtimes_3Q_8 % in TeX
G:=Group("D26:3Q8"); // GroupNames label
G:=SmallGroup(416,167);
// by ID
G=gap.SmallGroup(416,167);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,218,188,86,13829]);
// Polycyclic
G:=Group<a,b,c,d|a^26=b^2=c^4=1,d^2=c^2,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=c^-1>;
// generators/relations