direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C26, C10⋊C26, C130⋊3C2, C65⋊4C22, C5⋊(C2×C26), SmallGroup(260,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C26 |
Generators and relations for D5×C26
G = < a,b,c | a26=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C26
►On 130 points
Generators in S130
(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)
(1 93 45 117 78)(2 94 46 118 53)(3 95 47 119 54)(4 96 48 120 55)(5 97 49 121 56)(6 98 50 122 57)(7 99 51 123 58)(8 100 52 124 59)(9 101 27 125 60)(10 102 28 126 61)(11 103 29 127 62)(12 104 30 128 63)(13 79 31 129 64)(14 80 32 130 65)(15 81 33 105 66)(16 82 34 106 67)(17 83 35 107 68)(18 84 36 108 69)(19 85 37 109 70)(20 86 38 110 71)(21 87 39 111 72)(22 88 40 112 73)(23 89 41 113 74)(24 90 42 114 75)(25 91 43 115 76)(26 92 44 116 77)
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 73)(10 74)(11 75)(12 76)(13 77)(14 78)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 63)(26 64)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 51)(39 52)(79 116)(80 117)(81 118)(82 119)(83 120)(84 121)(85 122)(86 123)(87 124)(88 125)(89 126)(90 127)(91 128)(92 129)(93 130)(94 105)(95 106)(96 107)(97 108)(98 109)(99 110)(100 111)(101 112)(102 113)(103 114)(104 115)
G:=sub<Sym(130)| (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), (1,93,45,117,78)(2,94,46,118,53)(3,95,47,119,54)(4,96,48,120,55)(5,97,49,121,56)(6,98,50,122,57)(7,99,51,123,58)(8,100,52,124,59)(9,101,27,125,60)(10,102,28,126,61)(11,103,29,127,62)(12,104,30,128,63)(13,79,31,129,64)(14,80,32,130,65)(15,81,33,105,66)(16,82,34,106,67)(17,83,35,107,68)(18,84,36,108,69)(19,85,37,109,70)(20,86,38,110,71)(21,87,39,111,72)(22,88,40,112,73)(23,89,41,113,74)(24,90,42,114,75)(25,91,43,115,76)(26,92,44,116,77), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52)(79,116)(80,117)(81,118)(82,119)(83,120)(84,121)(85,122)(86,123)(87,124)(88,125)(89,126)(90,127)(91,128)(92,129)(93,130)(94,105)(95,106)(96,107)(97,108)(98,109)(99,110)(100,111)(101,112)(102,113)(103,114)(104,115)>;
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), (1,93,45,117,78)(2,94,46,118,53)(3,95,47,119,54)(4,96,48,120,55)(5,97,49,121,56)(6,98,50,122,57)(7,99,51,123,58)(8,100,52,124,59)(9,101,27,125,60)(10,102,28,126,61)(11,103,29,127,62)(12,104,30,128,63)(13,79,31,129,64)(14,80,32,130,65)(15,81,33,105,66)(16,82,34,106,67)(17,83,35,107,68)(18,84,36,108,69)(19,85,37,109,70)(20,86,38,110,71)(21,87,39,111,72)(22,88,40,112,73)(23,89,41,113,74)(24,90,42,114,75)(25,91,43,115,76)(26,92,44,116,77), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52)(79,116)(80,117)(81,118)(82,119)(83,120)(84,121)(85,122)(86,123)(87,124)(88,125)(89,126)(90,127)(91,128)(92,129)(93,130)(94,105)(95,106)(96,107)(97,108)(98,109)(99,110)(100,111)(101,112)(102,113)(103,114)(104,115) );
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)], [(1,93,45,117,78),(2,94,46,118,53),(3,95,47,119,54),(4,96,48,120,55),(5,97,49,121,56),(6,98,50,122,57),(7,99,51,123,58),(8,100,52,124,59),(9,101,27,125,60),(10,102,28,126,61),(11,103,29,127,62),(12,104,30,128,63),(13,79,31,129,64),(14,80,32,130,65),(15,81,33,105,66),(16,82,34,106,67),(17,83,35,107,68),(18,84,36,108,69),(19,85,37,109,70),(20,86,38,110,71),(21,87,39,111,72),(22,88,40,112,73),(23,89,41,113,74),(24,90,42,114,75),(25,91,43,115,76),(26,92,44,116,77)], [(1,65),(2,66),(3,67),(4,68),(5,69),(6,70),(7,71),(8,72),(9,73),(10,74),(11,75),(12,76),(13,77),(14,78),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(21,59),(22,60),(23,61),(24,62),(25,63),(26,64),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,49),(37,50),(38,51),(39,52),(79,116),(80,117),(81,118),(82,119),(83,120),(84,121),(85,122),(86,123),(87,124),(88,125),(89,126),(90,127),(91,128),(92,129),(93,130),(94,105),(95,106),(96,107),(97,108),(98,109),(99,110),(100,111),(101,112),(102,113),(103,114),(104,115)]])
104 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | 5B | 10A | 10B | 13A | ··· | 13L | 26A | ··· | 26L | 26M | ··· | 26AJ | 65A | ··· | 65X | 130A | ··· | 130X |
| order | 1 | 2 | 2 | 2 | 5 | 5 | 10 | 10 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 65 | ··· | 65 | 130 | ··· | 130 |
| size | 1 | 1 | 5 | 5 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C13 | C26 | C26 | D5 | D10 | D5×C13 | D5×C26 |
| kernel | D5×C26 | D5×C13 | C130 | D10 | D5 | C10 | C26 | C13 | C2 | C1 |
| # reps | 1 | 2 | 1 | 12 | 24 | 12 | 2 | 2 | 24 | 24 |
Matrix representation of D5×C26 ►in GL3(𝔽131) generated by
| 130 | 0 | 0 |
| 0 | 39 | 0 |
| 0 | 0 | 39 |
| 1 | 0 | 0 |
| 0 | 130 | 1 |
| 0 | 118 | 12 |
| 1 | 0 | 0 |
| 0 | 130 | 0 |
| 0 | 118 | 1 |
G:=sub<GL(3,GF(131))| [130,0,0,0,39,0,0,0,39],[1,0,0,0,130,118,0,1,12],[1,0,0,0,130,118,0,0,1] >;
D5×C26 in GAP, Magma, Sage, TeX
D_5\times C_{26} % in TeX
G:=Group("D5xC26"); // GroupNames label
G:=SmallGroup(260,13);
// by ID
G=gap.SmallGroup(260,13);
# by ID
G:=PCGroup([4,-2,-2,-13,-5,3331]);
// Polycyclic
G:=Group<a,b,c|a^26=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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