direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q16×D13, C8.9D26, Q8.3D26, Dic52⋊5C2, D26.25D4, C52.8C23, C104.7C22, Dic13.9D4, Dic26.4C22, C13⋊2(C2×Q16), (C13×Q16)⋊2C2, (C8×D13).1C2, C13⋊Q16⋊3C2, C2.22(D4×D13), C26.34(C2×D4), (Q8×D13).1C2, C4.8(C22×D13), C13⋊2C8.7C22, (Q8×C13).3C22, (C4×D13).19C22, SmallGroup(416,138)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q16×D13
G = < a,b,c,d | a8=c13=d2=1, b2=a4, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 400 in 60 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, Q8, Q8, C13, C2×C8, Q16, Q16, C2×Q8, D13, C26, C2×Q16, Dic13, Dic13, C52, C52, D26, C13⋊2C8, C104, Dic26, Dic26, C4×D13, C4×D13, Q8×C13, C8×D13, Dic52, C13⋊Q16, C13×Q16, Q8×D13, Q16×D13
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, D13, C2×Q16, D26, C22×D13, D4×D13, Q16×D13
►On 208 points
Generators in S208
(1 194 33 173 15 202 52 157)(2 195 34 174 16 203 40 158)(3 183 35 175 17 204 41 159)(4 184 36 176 18 205 42 160)(5 185 37 177 19 206 43 161)(6 186 38 178 20 207 44 162)(7 187 39 179 21 208 45 163)(8 188 27 180 22 196 46 164)(9 189 28 181 23 197 47 165)(10 190 29 182 24 198 48 166)(11 191 30 170 25 199 49 167)(12 192 31 171 26 200 50 168)(13 193 32 172 14 201 51 169)(53 143 97 113 77 144 83 129)(54 131 98 114 78 145 84 130)(55 132 99 115 66 146 85 118)(56 133 100 116 67 147 86 119)(57 134 101 117 68 148 87 120)(58 135 102 105 69 149 88 121)(59 136 103 106 70 150 89 122)(60 137 104 107 71 151 90 123)(61 138 92 108 72 152 91 124)(62 139 93 109 73 153 79 125)(63 140 94 110 74 154 80 126)(64 141 95 111 75 155 81 127)(65 142 96 112 76 156 82 128)
(1 69 15 58)(2 70 16 59)(3 71 17 60)(4 72 18 61)(5 73 19 62)(6 74 20 63)(7 75 21 64)(8 76 22 65)(9 77 23 53)(10 78 24 54)(11 66 25 55)(12 67 26 56)(13 68 14 57)(27 96 46 82)(28 97 47 83)(29 98 48 84)(30 99 49 85)(31 100 50 86)(32 101 51 87)(33 102 52 88)(34 103 40 89)(35 104 41 90)(36 92 42 91)(37 93 43 79)(38 94 44 80)(39 95 45 81)(105 202 121 194)(106 203 122 195)(107 204 123 183)(108 205 124 184)(109 206 125 185)(110 207 126 186)(111 208 127 187)(112 196 128 188)(113 197 129 189)(114 198 130 190)(115 199 118 191)(116 200 119 192)(117 201 120 193)(131 166 145 182)(132 167 146 170)(133 168 147 171)(134 169 148 172)(135 157 149 173)(136 158 150 174)(137 159 151 175)(138 160 152 176)(139 161 153 177)(140 162 154 178)(141 163 155 179)(142 164 156 180)(143 165 144 181)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 15)(16 26)(17 25)(18 24)(19 23)(20 22)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(40 50)(41 49)(42 48)(43 47)(44 46)(51 52)(53 62)(54 61)(55 60)(56 59)(57 58)(63 65)(66 71)(67 70)(68 69)(72 78)(73 77)(74 76)(79 83)(80 82)(84 91)(85 90)(86 89)(87 88)(92 98)(93 97)(94 96)(99 104)(100 103)(101 102)(105 117)(106 116)(107 115)(108 114)(109 113)(110 112)(118 123)(119 122)(120 121)(124 130)(125 129)(126 128)(131 138)(132 137)(133 136)(134 135)(139 143)(140 142)(144 153)(145 152)(146 151)(147 150)(148 149)(154 156)(157 169)(158 168)(159 167)(160 166)(161 165)(162 164)(170 175)(171 174)(172 173)(176 182)(177 181)(178 180)(183 191)(184 190)(185 189)(186 188)(192 195)(193 194)(196 207)(197 206)(198 205)(199 204)(200 203)(201 202)
G:=sub<Sym(208)| (1,194,33,173,15,202,52,157)(2,195,34,174,16,203,40,158)(3,183,35,175,17,204,41,159)(4,184,36,176,18,205,42,160)(5,185,37,177,19,206,43,161)(6,186,38,178,20,207,44,162)(7,187,39,179,21,208,45,163)(8,188,27,180,22,196,46,164)(9,189,28,181,23,197,47,165)(10,190,29,182,24,198,48,166)(11,191,30,170,25,199,49,167)(12,192,31,171,26,200,50,168)(13,193,32,172,14,201,51,169)(53,143,97,113,77,144,83,129)(54,131,98,114,78,145,84,130)(55,132,99,115,66,146,85,118)(56,133,100,116,67,147,86,119)(57,134,101,117,68,148,87,120)(58,135,102,105,69,149,88,121)(59,136,103,106,70,150,89,122)(60,137,104,107,71,151,90,123)(61,138,92,108,72,152,91,124)(62,139,93,109,73,153,79,125)(63,140,94,110,74,154,80,126)(64,141,95,111,75,155,81,127)(65,142,96,112,76,156,82,128), (1,69,15,58)(2,70,16,59)(3,71,17,60)(4,72,18,61)(5,73,19,62)(6,74,20,63)(7,75,21,64)(8,76,22,65)(9,77,23,53)(10,78,24,54)(11,66,25,55)(12,67,26,56)(13,68,14,57)(27,96,46,82)(28,97,47,83)(29,98,48,84)(30,99,49,85)(31,100,50,86)(32,101,51,87)(33,102,52,88)(34,103,40,89)(35,104,41,90)(36,92,42,91)(37,93,43,79)(38,94,44,80)(39,95,45,81)(105,202,121,194)(106,203,122,195)(107,204,123,183)(108,205,124,184)(109,206,125,185)(110,207,126,186)(111,208,127,187)(112,196,128,188)(113,197,129,189)(114,198,130,190)(115,199,118,191)(116,200,119,192)(117,201,120,193)(131,166,145,182)(132,167,146,170)(133,168,147,171)(134,169,148,172)(135,157,149,173)(136,158,150,174)(137,159,151,175)(138,160,152,176)(139,161,153,177)(140,162,154,178)(141,163,155,179)(142,164,156,180)(143,165,144,181), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,15)(16,26)(17,25)(18,24)(19,23)(20,22)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(40,50)(41,49)(42,48)(43,47)(44,46)(51,52)(53,62)(54,61)(55,60)(56,59)(57,58)(63,65)(66,71)(67,70)(68,69)(72,78)(73,77)(74,76)(79,83)(80,82)(84,91)(85,90)(86,89)(87,88)(92,98)(93,97)(94,96)(99,104)(100,103)(101,102)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112)(118,123)(119,122)(120,121)(124,130)(125,129)(126,128)(131,138)(132,137)(133,136)(134,135)(139,143)(140,142)(144,153)(145,152)(146,151)(147,150)(148,149)(154,156)(157,169)(158,168)(159,167)(160,166)(161,165)(162,164)(170,175)(171,174)(172,173)(176,182)(177,181)(178,180)(183,191)(184,190)(185,189)(186,188)(192,195)(193,194)(196,207)(197,206)(198,205)(199,204)(200,203)(201,202)>;
G:=Group( (1,194,33,173,15,202,52,157)(2,195,34,174,16,203,40,158)(3,183,35,175,17,204,41,159)(4,184,36,176,18,205,42,160)(5,185,37,177,19,206,43,161)(6,186,38,178,20,207,44,162)(7,187,39,179,21,208,45,163)(8,188,27,180,22,196,46,164)(9,189,28,181,23,197,47,165)(10,190,29,182,24,198,48,166)(11,191,30,170,25,199,49,167)(12,192,31,171,26,200,50,168)(13,193,32,172,14,201,51,169)(53,143,97,113,77,144,83,129)(54,131,98,114,78,145,84,130)(55,132,99,115,66,146,85,118)(56,133,100,116,67,147,86,119)(57,134,101,117,68,148,87,120)(58,135,102,105,69,149,88,121)(59,136,103,106,70,150,89,122)(60,137,104,107,71,151,90,123)(61,138,92,108,72,152,91,124)(62,139,93,109,73,153,79,125)(63,140,94,110,74,154,80,126)(64,141,95,111,75,155,81,127)(65,142,96,112,76,156,82,128), (1,69,15,58)(2,70,16,59)(3,71,17,60)(4,72,18,61)(5,73,19,62)(6,74,20,63)(7,75,21,64)(8,76,22,65)(9,77,23,53)(10,78,24,54)(11,66,25,55)(12,67,26,56)(13,68,14,57)(27,96,46,82)(28,97,47,83)(29,98,48,84)(30,99,49,85)(31,100,50,86)(32,101,51,87)(33,102,52,88)(34,103,40,89)(35,104,41,90)(36,92,42,91)(37,93,43,79)(38,94,44,80)(39,95,45,81)(105,202,121,194)(106,203,122,195)(107,204,123,183)(108,205,124,184)(109,206,125,185)(110,207,126,186)(111,208,127,187)(112,196,128,188)(113,197,129,189)(114,198,130,190)(115,199,118,191)(116,200,119,192)(117,201,120,193)(131,166,145,182)(132,167,146,170)(133,168,147,171)(134,169,148,172)(135,157,149,173)(136,158,150,174)(137,159,151,175)(138,160,152,176)(139,161,153,177)(140,162,154,178)(141,163,155,179)(142,164,156,180)(143,165,144,181), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,15)(16,26)(17,25)(18,24)(19,23)(20,22)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(40,50)(41,49)(42,48)(43,47)(44,46)(51,52)(53,62)(54,61)(55,60)(56,59)(57,58)(63,65)(66,71)(67,70)(68,69)(72,78)(73,77)(74,76)(79,83)(80,82)(84,91)(85,90)(86,89)(87,88)(92,98)(93,97)(94,96)(99,104)(100,103)(101,102)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112)(118,123)(119,122)(120,121)(124,130)(125,129)(126,128)(131,138)(132,137)(133,136)(134,135)(139,143)(140,142)(144,153)(145,152)(146,151)(147,150)(148,149)(154,156)(157,169)(158,168)(159,167)(160,166)(161,165)(162,164)(170,175)(171,174)(172,173)(176,182)(177,181)(178,180)(183,191)(184,190)(185,189)(186,188)(192,195)(193,194)(196,207)(197,206)(198,205)(199,204)(200,203)(201,202) );
G=PermutationGroup([[(1,194,33,173,15,202,52,157),(2,195,34,174,16,203,40,158),(3,183,35,175,17,204,41,159),(4,184,36,176,18,205,42,160),(5,185,37,177,19,206,43,161),(6,186,38,178,20,207,44,162),(7,187,39,179,21,208,45,163),(8,188,27,180,22,196,46,164),(9,189,28,181,23,197,47,165),(10,190,29,182,24,198,48,166),(11,191,30,170,25,199,49,167),(12,192,31,171,26,200,50,168),(13,193,32,172,14,201,51,169),(53,143,97,113,77,144,83,129),(54,131,98,114,78,145,84,130),(55,132,99,115,66,146,85,118),(56,133,100,116,67,147,86,119),(57,134,101,117,68,148,87,120),(58,135,102,105,69,149,88,121),(59,136,103,106,70,150,89,122),(60,137,104,107,71,151,90,123),(61,138,92,108,72,152,91,124),(62,139,93,109,73,153,79,125),(63,140,94,110,74,154,80,126),(64,141,95,111,75,155,81,127),(65,142,96,112,76,156,82,128)], [(1,69,15,58),(2,70,16,59),(3,71,17,60),(4,72,18,61),(5,73,19,62),(6,74,20,63),(7,75,21,64),(8,76,22,65),(9,77,23,53),(10,78,24,54),(11,66,25,55),(12,67,26,56),(13,68,14,57),(27,96,46,82),(28,97,47,83),(29,98,48,84),(30,99,49,85),(31,100,50,86),(32,101,51,87),(33,102,52,88),(34,103,40,89),(35,104,41,90),(36,92,42,91),(37,93,43,79),(38,94,44,80),(39,95,45,81),(105,202,121,194),(106,203,122,195),(107,204,123,183),(108,205,124,184),(109,206,125,185),(110,207,126,186),(111,208,127,187),(112,196,128,188),(113,197,129,189),(114,198,130,190),(115,199,118,191),(116,200,119,192),(117,201,120,193),(131,166,145,182),(132,167,146,170),(133,168,147,171),(134,169,148,172),(135,157,149,173),(136,158,150,174),(137,159,151,175),(138,160,152,176),(139,161,153,177),(140,162,154,178),(141,163,155,179),(142,164,156,180),(143,165,144,181)], [(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)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,15),(16,26),(17,25),(18,24),(19,23),(20,22),(27,38),(28,37),(29,36),(30,35),(31,34),(32,33),(40,50),(41,49),(42,48),(43,47),(44,46),(51,52),(53,62),(54,61),(55,60),(56,59),(57,58),(63,65),(66,71),(67,70),(68,69),(72,78),(73,77),(74,76),(79,83),(80,82),(84,91),(85,90),(86,89),(87,88),(92,98),(93,97),(94,96),(99,104),(100,103),(101,102),(105,117),(106,116),(107,115),(108,114),(109,113),(110,112),(118,123),(119,122),(120,121),(124,130),(125,129),(126,128),(131,138),(132,137),(133,136),(134,135),(139,143),(140,142),(144,153),(145,152),(146,151),(147,150),(148,149),(154,156),(157,169),(158,168),(159,167),(160,166),(161,165),(162,164),(170,175),(171,174),(172,173),(176,182),(177,181),(178,180),(183,191),(184,190),(185,189),(186,188),(192,195),(193,194),(196,207),(197,206),(198,205),(199,204),(200,203),(201,202)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 13A | ··· | 13F | 26A | ··· | 26F | 52A | ··· | 52F | 52G | ··· | 52R | 104A | ··· | 104L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 13 | 13 | 2 | 4 | 4 | 26 | 52 | 52 | 2 | 2 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
56 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 | D4 | Q16 | D13 | D26 | D26 | D4×D13 | Q16×D13 |
| kernel | Q16×D13 | C8×D13 | Dic52 | C13⋊Q16 | C13×Q16 | Q8×D13 | Dic13 | D26 | D13 | Q16 | C8 | Q8 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 6 | 6 | 12 | 6 | 12 |
Matrix representation of Q16×D13 ►in GL4(𝔽313) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 179 |
| 0 | 0 | 306 | 120 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 154 | 284 |
| 0 | 0 | 235 | 159 |
| 198 | 1 | 0 | 0 |
| 312 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 91 | 235 | 0 | 0 |
| 58 | 222 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(313))| [1,0,0,0,0,1,0,0,0,0,0,306,0,0,179,120],[1,0,0,0,0,1,0,0,0,0,154,235,0,0,284,159],[198,312,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[91,58,0,0,235,222,0,0,0,0,1,0,0,0,0,1] >;
Q16×D13 in GAP, Magma, Sage, TeX
Q_{16}\times D_{13} % in TeX
G:=Group("Q16xD13"); // GroupNames label
G:=SmallGroup(416,138);
// by ID
G=gap.SmallGroup(416,138);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,116,86,297,159,69,13829]);
// Polycyclic
G:=Group<a,b,c,d|a^8=c^13=d^2=1,b^2=a^4,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations