direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C7×D19, C19⋊C14, C133⋊2C2, SmallGroup(266,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C7×D19 |
Generators and relations for C7×D19
G = < a,b,c | a7=b19=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C7×D19
►On 133 points
Generators in S133
(1 132 101 84 70 43 37)(2 133 102 85 71 44 38)(3 115 103 86 72 45 20)(4 116 104 87 73 46 21)(5 117 105 88 74 47 22)(6 118 106 89 75 48 23)(7 119 107 90 76 49 24)(8 120 108 91 58 50 25)(9 121 109 92 59 51 26)(10 122 110 93 60 52 27)(11 123 111 94 61 53 28)(12 124 112 95 62 54 29)(13 125 113 77 63 55 30)(14 126 114 78 64 56 31)(15 127 96 79 65 57 32)(16 128 97 80 66 39 33)(17 129 98 81 67 40 34)(18 130 99 82 68 41 35)(19 131 100 83 69 42 36)
(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)
(1 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 34)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(35 38)(36 37)(39 46)(40 45)(41 44)(42 43)(47 57)(48 56)(49 55)(50 54)(51 53)(58 62)(59 61)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 70)(77 90)(78 89)(79 88)(80 87)(81 86)(82 85)(83 84)(91 95)(92 94)(96 105)(97 104)(98 103)(99 102)(100 101)(106 114)(107 113)(108 112)(109 111)(115 129)(116 128)(117 127)(118 126)(119 125)(120 124)(121 123)(130 133)(131 132)
G:=sub<Sym(133)| (1,132,101,84,70,43,37)(2,133,102,85,71,44,38)(3,115,103,86,72,45,20)(4,116,104,87,73,46,21)(5,117,105,88,74,47,22)(6,118,106,89,75,48,23)(7,119,107,90,76,49,24)(8,120,108,91,58,50,25)(9,121,109,92,59,51,26)(10,122,110,93,60,52,27)(11,123,111,94,61,53,28)(12,124,112,95,62,54,29)(13,125,113,77,63,55,30)(14,126,114,78,64,56,31)(15,127,96,79,65,57,32)(16,128,97,80,66,39,33)(17,129,98,81,67,40,34)(18,130,99,82,68,41,35)(19,131,100,83,69,42,36), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(35,38)(36,37)(39,46)(40,45)(41,44)(42,43)(47,57)(48,56)(49,55)(50,54)(51,53)(58,62)(59,61)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84)(91,95)(92,94)(96,105)(97,104)(98,103)(99,102)(100,101)(106,114)(107,113)(108,112)(109,111)(115,129)(116,128)(117,127)(118,126)(119,125)(120,124)(121,123)(130,133)(131,132)>;
G:=Group( (1,132,101,84,70,43,37)(2,133,102,85,71,44,38)(3,115,103,86,72,45,20)(4,116,104,87,73,46,21)(5,117,105,88,74,47,22)(6,118,106,89,75,48,23)(7,119,107,90,76,49,24)(8,120,108,91,58,50,25)(9,121,109,92,59,51,26)(10,122,110,93,60,52,27)(11,123,111,94,61,53,28)(12,124,112,95,62,54,29)(13,125,113,77,63,55,30)(14,126,114,78,64,56,31)(15,127,96,79,65,57,32)(16,128,97,80,66,39,33)(17,129,98,81,67,40,34)(18,130,99,82,68,41,35)(19,131,100,83,69,42,36), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(35,38)(36,37)(39,46)(40,45)(41,44)(42,43)(47,57)(48,56)(49,55)(50,54)(51,53)(58,62)(59,61)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84)(91,95)(92,94)(96,105)(97,104)(98,103)(99,102)(100,101)(106,114)(107,113)(108,112)(109,111)(115,129)(116,128)(117,127)(118,126)(119,125)(120,124)(121,123)(130,133)(131,132) );
G=PermutationGroup([[(1,132,101,84,70,43,37),(2,133,102,85,71,44,38),(3,115,103,86,72,45,20),(4,116,104,87,73,46,21),(5,117,105,88,74,47,22),(6,118,106,89,75,48,23),(7,119,107,90,76,49,24),(8,120,108,91,58,50,25),(9,121,109,92,59,51,26),(10,122,110,93,60,52,27),(11,123,111,94,61,53,28),(12,124,112,95,62,54,29),(13,125,113,77,63,55,30),(14,126,114,78,64,56,31),(15,127,96,79,65,57,32),(16,128,97,80,66,39,33),(17,129,98,81,67,40,34),(18,130,99,82,68,41,35),(19,131,100,83,69,42,36)], [(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)], [(1,19),(2,18),(3,17),(4,16),(5,15),(6,14),(7,13),(8,12),(9,11),(20,34),(21,33),(22,32),(23,31),(24,30),(25,29),(26,28),(35,38),(36,37),(39,46),(40,45),(41,44),(42,43),(47,57),(48,56),(49,55),(50,54),(51,53),(58,62),(59,61),(63,76),(64,75),(65,74),(66,73),(67,72),(68,71),(69,70),(77,90),(78,89),(79,88),(80,87),(81,86),(82,85),(83,84),(91,95),(92,94),(96,105),(97,104),(98,103),(99,102),(100,101),(106,114),(107,113),(108,112),(109,111),(115,129),(116,128),(117,127),(118,126),(119,125),(120,124),(121,123),(130,133),(131,132)]])
77 conjugacy classes
| class | 1 | 2 | 7A | ··· | 7F | 14A | ··· | 14F | 19A | ··· | 19I | 133A | ··· | 133BB |
| order | 1 | 2 | 7 | ··· | 7 | 14 | ··· | 14 | 19 | ··· | 19 | 133 | ··· | 133 |
| size | 1 | 19 | 1 | ··· | 1 | 19 | ··· | 19 | 2 | ··· | 2 | 2 | ··· | 2 |
77 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C7 | C14 | D19 | C7×D19 |
| kernel | C7×D19 | C133 | D19 | C19 | C7 | C1 |
| # reps | 1 | 1 | 6 | 6 | 9 | 54 |
Matrix representation of C7×D19 ►in GL2(𝔽1597) generated by
| 1150 | 0 |
| 0 | 1150 |
| 159 | 1 |
| 1372 | 320 |
| 185 | 551 |
| 448 | 1412 |
G:=sub<GL(2,GF(1597))| [1150,0,0,1150],[159,1372,1,320],[185,448,551,1412] >;
C7×D19 in GAP, Magma, Sage, TeX
C_7\times D_{19} % in TeX
G:=Group("C7xD19"); // GroupNames label
G:=SmallGroup(266,2);
// by ID
G=gap.SmallGroup(266,2);
# by ID
G:=PCGroup([3,-2,-7,-19,2270]);
// Polycyclic
G:=Group<a,b,c|a^7=b^19=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export