direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D5×C29, C5⋊C58, C145⋊3C2, SmallGroup(290,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C29 |
Generators and relations for D5×C29
G = < a,b,c | a29=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C29
►On 145 points
Generators in S145
(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)
(1 140 77 36 91)(2 141 78 37 92)(3 142 79 38 93)(4 143 80 39 94)(5 144 81 40 95)(6 145 82 41 96)(7 117 83 42 97)(8 118 84 43 98)(9 119 85 44 99)(10 120 86 45 100)(11 121 87 46 101)(12 122 59 47 102)(13 123 60 48 103)(14 124 61 49 104)(15 125 62 50 105)(16 126 63 51 106)(17 127 64 52 107)(18 128 65 53 108)(19 129 66 54 109)(20 130 67 55 110)(21 131 68 56 111)(22 132 69 57 112)(23 133 70 58 113)(24 134 71 30 114)(25 135 72 31 115)(26 136 73 32 116)(27 137 74 33 88)(28 138 75 34 89)(29 139 76 35 90)
(1 91)(2 92)(3 93)(4 94)(5 95)(6 96)(7 97)(8 98)(9 99)(10 100)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 111)(22 112)(23 113)(24 114)(25 115)(26 116)(27 88)(28 89)(29 90)(30 134)(31 135)(32 136)(33 137)(34 138)(35 139)(36 140)(37 141)(38 142)(39 143)(40 144)(41 145)(42 117)(43 118)(44 119)(45 120)(46 121)(47 122)(48 123)(49 124)(50 125)(51 126)(52 127)(53 128)(54 129)(55 130)(56 131)(57 132)(58 133)
G:=sub<Sym(145)| (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), (1,140,77,36,91)(2,141,78,37,92)(3,142,79,38,93)(4,143,80,39,94)(5,144,81,40,95)(6,145,82,41,96)(7,117,83,42,97)(8,118,84,43,98)(9,119,85,44,99)(10,120,86,45,100)(11,121,87,46,101)(12,122,59,47,102)(13,123,60,48,103)(14,124,61,49,104)(15,125,62,50,105)(16,126,63,51,106)(17,127,64,52,107)(18,128,65,53,108)(19,129,66,54,109)(20,130,67,55,110)(21,131,68,56,111)(22,132,69,57,112)(23,133,70,58,113)(24,134,71,30,114)(25,135,72,31,115)(26,136,73,32,116)(27,137,74,33,88)(28,138,75,34,89)(29,139,76,35,90), (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,97)(8,98)(9,99)(10,100)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,111)(22,112)(23,113)(24,114)(25,115)(26,116)(27,88)(28,89)(29,90)(30,134)(31,135)(32,136)(33,137)(34,138)(35,139)(36,140)(37,141)(38,142)(39,143)(40,144)(41,145)(42,117)(43,118)(44,119)(45,120)(46,121)(47,122)(48,123)(49,124)(50,125)(51,126)(52,127)(53,128)(54,129)(55,130)(56,131)(57,132)(58,133)>;
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), (1,140,77,36,91)(2,141,78,37,92)(3,142,79,38,93)(4,143,80,39,94)(5,144,81,40,95)(6,145,82,41,96)(7,117,83,42,97)(8,118,84,43,98)(9,119,85,44,99)(10,120,86,45,100)(11,121,87,46,101)(12,122,59,47,102)(13,123,60,48,103)(14,124,61,49,104)(15,125,62,50,105)(16,126,63,51,106)(17,127,64,52,107)(18,128,65,53,108)(19,129,66,54,109)(20,130,67,55,110)(21,131,68,56,111)(22,132,69,57,112)(23,133,70,58,113)(24,134,71,30,114)(25,135,72,31,115)(26,136,73,32,116)(27,137,74,33,88)(28,138,75,34,89)(29,139,76,35,90), (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,97)(8,98)(9,99)(10,100)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,111)(22,112)(23,113)(24,114)(25,115)(26,116)(27,88)(28,89)(29,90)(30,134)(31,135)(32,136)(33,137)(34,138)(35,139)(36,140)(37,141)(38,142)(39,143)(40,144)(41,145)(42,117)(43,118)(44,119)(45,120)(46,121)(47,122)(48,123)(49,124)(50,125)(51,126)(52,127)(53,128)(54,129)(55,130)(56,131)(57,132)(58,133) );
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)], [(1,140,77,36,91),(2,141,78,37,92),(3,142,79,38,93),(4,143,80,39,94),(5,144,81,40,95),(6,145,82,41,96),(7,117,83,42,97),(8,118,84,43,98),(9,119,85,44,99),(10,120,86,45,100),(11,121,87,46,101),(12,122,59,47,102),(13,123,60,48,103),(14,124,61,49,104),(15,125,62,50,105),(16,126,63,51,106),(17,127,64,52,107),(18,128,65,53,108),(19,129,66,54,109),(20,130,67,55,110),(21,131,68,56,111),(22,132,69,57,112),(23,133,70,58,113),(24,134,71,30,114),(25,135,72,31,115),(26,136,73,32,116),(27,137,74,33,88),(28,138,75,34,89),(29,139,76,35,90)], [(1,91),(2,92),(3,93),(4,94),(5,95),(6,96),(7,97),(8,98),(9,99),(10,100),(11,101),(12,102),(13,103),(14,104),(15,105),(16,106),(17,107),(18,108),(19,109),(20,110),(21,111),(22,112),(23,113),(24,114),(25,115),(26,116),(27,88),(28,89),(29,90),(30,134),(31,135),(32,136),(33,137),(34,138),(35,139),(36,140),(37,141),(38,142),(39,143),(40,144),(41,145),(42,117),(43,118),(44,119),(45,120),(46,121),(47,122),(48,123),(49,124),(50,125),(51,126),(52,127),(53,128),(54,129),(55,130),(56,131),(57,132),(58,133)]])
116 conjugacy classes
| class | 1 | 2 | 5A | 5B | 29A | ··· | 29AB | 58A | ··· | 58AB | 145A | ··· | 145BD |
| order | 1 | 2 | 5 | 5 | 29 | ··· | 29 | 58 | ··· | 58 | 145 | ··· | 145 |
| size | 1 | 5 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 |
116 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C29 | C58 | D5 | D5×C29 |
| kernel | D5×C29 | C145 | D5 | C5 | C29 | C1 |
| # reps | 1 | 1 | 28 | 28 | 2 | 56 |
Matrix representation of D5×C29 ►in GL2(𝔽1451) generated by
| 686 | 0 |
| 0 | 686 |
| 1169 | 1170 |
| 1450 | 1450 |
| 1450 | 281 |
| 0 | 1 |
G:=sub<GL(2,GF(1451))| [686,0,0,686],[1169,1450,1170,1450],[1450,0,281,1] >;
D5×C29 in GAP, Magma, Sage, TeX
D_5\times C_{29} % in TeX
G:=Group("D5xC29"); // GroupNames label
G:=SmallGroup(290,1);
// by ID
G=gap.SmallGroup(290,1);
# by ID
G:=PCGroup([3,-2,-29,-5,2090]);
// Polycyclic
G:=Group<a,b,c|a^29=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export