direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×D49, C49⋊3C6, C147⋊2C2, C21.2D7, C7.(C3×D7), SmallGroup(294,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C49 — C3×D49 |
Generators and relations for C3×D49
G = < a,b,c | a3=b49=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D49
►On 147 points
Generators in S147
(1 100 87)(2 101 88)(3 102 89)(4 103 90)(5 104 91)(6 105 92)(7 106 93)(8 107 94)(9 108 95)(10 109 96)(11 110 97)(12 111 98)(13 112 50)(14 113 51)(15 114 52)(16 115 53)(17 116 54)(18 117 55)(19 118 56)(20 119 57)(21 120 58)(22 121 59)(23 122 60)(24 123 61)(25 124 62)(26 125 63)(27 126 64)(28 127 65)(29 128 66)(30 129 67)(31 130 68)(32 131 69)(33 132 70)(34 133 71)(35 134 72)(36 135 73)(37 136 74)(38 137 75)(39 138 76)(40 139 77)(41 140 78)(42 141 79)(43 142 80)(44 143 81)(45 144 82)(46 145 83)(47 146 84)(48 147 85)(49 99 86)
(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)
(1 49)(2 48)(3 47)(4 46)(5 45)(6 44)(7 43)(8 42)(9 41)(10 40)(11 39)(12 38)(13 37)(14 36)(15 35)(16 34)(17 33)(18 32)(19 31)(20 30)(21 29)(22 28)(23 27)(24 26)(50 74)(51 73)(52 72)(53 71)(54 70)(55 69)(56 68)(57 67)(58 66)(59 65)(60 64)(61 63)(75 98)(76 97)(77 96)(78 95)(79 94)(80 93)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(99 100)(101 147)(102 146)(103 145)(104 144)(105 143)(106 142)(107 141)(108 140)(109 139)(110 138)(111 137)(112 136)(113 135)(114 134)(115 133)(116 132)(117 131)(118 130)(119 129)(120 128)(121 127)(122 126)(123 125)
G:=sub<Sym(147)| (1,100,87)(2,101,88)(3,102,89)(4,103,90)(5,104,91)(6,105,92)(7,106,93)(8,107,94)(9,108,95)(10,109,96)(11,110,97)(12,111,98)(13,112,50)(14,113,51)(15,114,52)(16,115,53)(17,116,54)(18,117,55)(19,118,56)(20,119,57)(21,120,58)(22,121,59)(23,122,60)(24,123,61)(25,124,62)(26,125,63)(27,126,64)(28,127,65)(29,128,66)(30,129,67)(31,130,68)(32,131,69)(33,132,70)(34,133,71)(35,134,72)(36,135,73)(37,136,74)(38,137,75)(39,138,76)(40,139,77)(41,140,78)(42,141,79)(43,142,80)(44,143,81)(45,144,82)(46,145,83)(47,146,84)(48,147,85)(49,99,86), (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), (1,49)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(17,33)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)(50,74)(51,73)(52,72)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64)(61,63)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(99,100)(101,147)(102,146)(103,145)(104,144)(105,143)(106,142)(107,141)(108,140)(109,139)(110,138)(111,137)(112,136)(113,135)(114,134)(115,133)(116,132)(117,131)(118,130)(119,129)(120,128)(121,127)(122,126)(123,125)>;
G:=Group( (1,100,87)(2,101,88)(3,102,89)(4,103,90)(5,104,91)(6,105,92)(7,106,93)(8,107,94)(9,108,95)(10,109,96)(11,110,97)(12,111,98)(13,112,50)(14,113,51)(15,114,52)(16,115,53)(17,116,54)(18,117,55)(19,118,56)(20,119,57)(21,120,58)(22,121,59)(23,122,60)(24,123,61)(25,124,62)(26,125,63)(27,126,64)(28,127,65)(29,128,66)(30,129,67)(31,130,68)(32,131,69)(33,132,70)(34,133,71)(35,134,72)(36,135,73)(37,136,74)(38,137,75)(39,138,76)(40,139,77)(41,140,78)(42,141,79)(43,142,80)(44,143,81)(45,144,82)(46,145,83)(47,146,84)(48,147,85)(49,99,86), (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), (1,49)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(17,33)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)(50,74)(51,73)(52,72)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64)(61,63)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(99,100)(101,147)(102,146)(103,145)(104,144)(105,143)(106,142)(107,141)(108,140)(109,139)(110,138)(111,137)(112,136)(113,135)(114,134)(115,133)(116,132)(117,131)(118,130)(119,129)(120,128)(121,127)(122,126)(123,125) );
G=PermutationGroup([[(1,100,87),(2,101,88),(3,102,89),(4,103,90),(5,104,91),(6,105,92),(7,106,93),(8,107,94),(9,108,95),(10,109,96),(11,110,97),(12,111,98),(13,112,50),(14,113,51),(15,114,52),(16,115,53),(17,116,54),(18,117,55),(19,118,56),(20,119,57),(21,120,58),(22,121,59),(23,122,60),(24,123,61),(25,124,62),(26,125,63),(27,126,64),(28,127,65),(29,128,66),(30,129,67),(31,130,68),(32,131,69),(33,132,70),(34,133,71),(35,134,72),(36,135,73),(37,136,74),(38,137,75),(39,138,76),(40,139,77),(41,140,78),(42,141,79),(43,142,80),(44,143,81),(45,144,82),(46,145,83),(47,146,84),(48,147,85),(49,99,86)], [(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)], [(1,49),(2,48),(3,47),(4,46),(5,45),(6,44),(7,43),(8,42),(9,41),(10,40),(11,39),(12,38),(13,37),(14,36),(15,35),(16,34),(17,33),(18,32),(19,31),(20,30),(21,29),(22,28),(23,27),(24,26),(50,74),(51,73),(52,72),(53,71),(54,70),(55,69),(56,68),(57,67),(58,66),(59,65),(60,64),(61,63),(75,98),(76,97),(77,96),(78,95),(79,94),(80,93),(81,92),(82,91),(83,90),(84,89),(85,88),(86,87),(99,100),(101,147),(102,146),(103,145),(104,144),(105,143),(106,142),(107,141),(108,140),(109,139),(110,138),(111,137),(112,136),(113,135),(114,134),(115,133),(116,132),(117,131),(118,130),(119,129),(120,128),(121,127),(122,126),(123,125)]])
78 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7A | 7B | 7C | 21A | ··· | 21F | 49A | ··· | 49U | 147A | ··· | 147AP |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 7 | 7 | 7 | 21 | ··· | 21 | 49 | ··· | 49 | 147 | ··· | 147 |
| size | 1 | 49 | 1 | 1 | 49 | 49 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C3 | C6 | D7 | C3×D7 | D49 | C3×D49 |
| kernel | C3×D49 | C147 | D49 | C49 | C21 | C7 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 3 | 6 | 21 | 42 |
Matrix representation of C3×D49 ►in GL3(𝔽883) generated by
| 337 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 784 | 406 |
| 0 | 71 | 351 |
| 882 | 0 | 0 |
| 0 | 130 | 337 |
| 0 | 618 | 753 |
G:=sub<GL(3,GF(883))| [337,0,0,0,1,0,0,0,1],[1,0,0,0,784,71,0,406,351],[882,0,0,0,130,618,0,337,753] >;
C3×D49 in GAP, Magma, Sage, TeX
C_3\times D_{49} % in TeX
G:=Group("C3xD49"); // GroupNames label
G:=SmallGroup(294,4);
// by ID
G=gap.SmallGroup(294,4);
# by ID
G:=PCGroup([4,-2,-3,-7,-7,938,514,4035]);
// Polycyclic
G:=Group<a,b,c|a^3=b^49=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export