direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×D29, C29⋊C10, C145⋊2C2, SmallGroup(290,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C29 — C5×D29 |
Generators and relations for C5×D29
G = < a,b,c | a5=b29=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C5×D29
►On 145 points
Generators in S145
(1 127 112 65 44)(2 128 113 66 45)(3 129 114 67 46)(4 130 115 68 47)(5 131 116 69 48)(6 132 88 70 49)(7 133 89 71 50)(8 134 90 72 51)(9 135 91 73 52)(10 136 92 74 53)(11 137 93 75 54)(12 138 94 76 55)(13 139 95 77 56)(14 140 96 78 57)(15 141 97 79 58)(16 142 98 80 30)(17 143 99 81 31)(18 144 100 82 32)(19 145 101 83 33)(20 117 102 84 34)(21 118 103 85 35)(22 119 104 86 36)(23 120 105 87 37)(24 121 106 59 38)(25 122 107 60 39)(26 123 108 61 40)(27 124 109 62 41)(28 125 110 63 42)(29 126 111 64 43)
(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 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(13 17)(14 16)(30 57)(31 56)(32 55)(33 54)(34 53)(35 52)(36 51)(37 50)(38 49)(39 48)(40 47)(41 46)(42 45)(43 44)(59 70)(60 69)(61 68)(62 67)(63 66)(64 65)(71 87)(72 86)(73 85)(74 84)(75 83)(76 82)(77 81)(78 80)(88 106)(89 105)(90 104)(91 103)(92 102)(93 101)(94 100)(95 99)(96 98)(107 116)(108 115)(109 114)(110 113)(111 112)(117 136)(118 135)(119 134)(120 133)(121 132)(122 131)(123 130)(124 129)(125 128)(126 127)(137 145)(138 144)(139 143)(140 142)
G:=sub<Sym(145)| (1,127,112,65,44)(2,128,113,66,45)(3,129,114,67,46)(4,130,115,68,47)(5,131,116,69,48)(6,132,88,70,49)(7,133,89,71,50)(8,134,90,72,51)(9,135,91,73,52)(10,136,92,74,53)(11,137,93,75,54)(12,138,94,76,55)(13,139,95,77,56)(14,140,96,78,57)(15,141,97,79,58)(16,142,98,80,30)(17,143,99,81,31)(18,144,100,82,32)(19,145,101,83,33)(20,117,102,84,34)(21,118,103,85,35)(22,119,104,86,36)(23,120,105,87,37)(24,121,106,59,38)(25,122,107,60,39)(26,123,108,61,40)(27,124,109,62,41)(28,125,110,63,42)(29,126,111,64,43), (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)(71,87)(72,86)(73,85)(74,84)(75,83)(76,82)(77,81)(78,80)(88,106)(89,105)(90,104)(91,103)(92,102)(93,101)(94,100)(95,99)(96,98)(107,116)(108,115)(109,114)(110,113)(111,112)(117,136)(118,135)(119,134)(120,133)(121,132)(122,131)(123,130)(124,129)(125,128)(126,127)(137,145)(138,144)(139,143)(140,142)>;
G:=Group( (1,127,112,65,44)(2,128,113,66,45)(3,129,114,67,46)(4,130,115,68,47)(5,131,116,69,48)(6,132,88,70,49)(7,133,89,71,50)(8,134,90,72,51)(9,135,91,73,52)(10,136,92,74,53)(11,137,93,75,54)(12,138,94,76,55)(13,139,95,77,56)(14,140,96,78,57)(15,141,97,79,58)(16,142,98,80,30)(17,143,99,81,31)(18,144,100,82,32)(19,145,101,83,33)(20,117,102,84,34)(21,118,103,85,35)(22,119,104,86,36)(23,120,105,87,37)(24,121,106,59,38)(25,122,107,60,39)(26,123,108,61,40)(27,124,109,62,41)(28,125,110,63,42)(29,126,111,64,43), (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)(71,87)(72,86)(73,85)(74,84)(75,83)(76,82)(77,81)(78,80)(88,106)(89,105)(90,104)(91,103)(92,102)(93,101)(94,100)(95,99)(96,98)(107,116)(108,115)(109,114)(110,113)(111,112)(117,136)(118,135)(119,134)(120,133)(121,132)(122,131)(123,130)(124,129)(125,128)(126,127)(137,145)(138,144)(139,143)(140,142) );
G=PermutationGroup([[(1,127,112,65,44),(2,128,113,66,45),(3,129,114,67,46),(4,130,115,68,47),(5,131,116,69,48),(6,132,88,70,49),(7,133,89,71,50),(8,134,90,72,51),(9,135,91,73,52),(10,136,92,74,53),(11,137,93,75,54),(12,138,94,76,55),(13,139,95,77,56),(14,140,96,78,57),(15,141,97,79,58),(16,142,98,80,30),(17,143,99,81,31),(18,144,100,82,32),(19,145,101,83,33),(20,117,102,84,34),(21,118,103,85,35),(22,119,104,86,36),(23,120,105,87,37),(24,121,106,59,38),(25,122,107,60,39),(26,123,108,61,40),(27,124,109,62,41),(28,125,110,63,42),(29,126,111,64,43)], [(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)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,20),(11,19),(12,18),(13,17),(14,16),(30,57),(31,56),(32,55),(33,54),(34,53),(35,52),(36,51),(37,50),(38,49),(39,48),(40,47),(41,46),(42,45),(43,44),(59,70),(60,69),(61,68),(62,67),(63,66),(64,65),(71,87),(72,86),(73,85),(74,84),(75,83),(76,82),(77,81),(78,80),(88,106),(89,105),(90,104),(91,103),(92,102),(93,101),(94,100),(95,99),(96,98),(107,116),(108,115),(109,114),(110,113),(111,112),(117,136),(118,135),(119,134),(120,133),(121,132),(122,131),(123,130),(124,129),(125,128),(126,127),(137,145),(138,144),(139,143),(140,142)]])
80 conjugacy classes
| class | 1 | 2 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 29A | ··· | 29N | 145A | ··· | 145BD |
| order | 1 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 29 | ··· | 29 | 145 | ··· | 145 |
| size | 1 | 29 | 1 | 1 | 1 | 1 | 29 | 29 | 29 | 29 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C5 | C10 | D29 | C5×D29 |
| kernel | C5×D29 | C145 | D29 | C29 | C5 | C1 |
| # reps | 1 | 1 | 4 | 4 | 14 | 56 |
Matrix representation of C5×D29 ►in GL2(𝔽1451) generated by
| 545 | 0 |
| 0 | 545 |
| 0 | 1 |
| 1450 | 127 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(1451))| [545,0,0,545],[0,1450,1,127],[0,1,1,0] >;
C5×D29 in GAP, Magma, Sage, TeX
C_5\times D_{29} % in TeX
G:=Group("C5xD29"); // GroupNames label
G:=SmallGroup(290,2);
// by ID
G=gap.SmallGroup(290,2);
# by ID
G:=PCGroup([3,-2,-5,-29,2522]);
// Polycyclic
G:=Group<a,b,c|a^5=b^29=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export