direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D7×C19, C7⋊C38, C133⋊3C2, SmallGroup(266,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C19 |
Generators and relations for D7×C19
G = < a,b,c | a19=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D7×C19
►On 133 points
Generators in S133
(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 49 66 113 77 130 23)(2 50 67 114 78 131 24)(3 51 68 96 79 132 25)(4 52 69 97 80 133 26)(5 53 70 98 81 115 27)(6 54 71 99 82 116 28)(7 55 72 100 83 117 29)(8 56 73 101 84 118 30)(9 57 74 102 85 119 31)(10 39 75 103 86 120 32)(11 40 76 104 87 121 33)(12 41 58 105 88 122 34)(13 42 59 106 89 123 35)(14 43 60 107 90 124 36)(15 44 61 108 91 125 37)(16 45 62 109 92 126 38)(17 46 63 110 93 127 20)(18 47 64 111 94 128 21)(19 48 65 112 95 129 22)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 20)(18 21)(19 22)(39 120)(40 121)(41 122)(42 123)(43 124)(44 125)(45 126)(46 127)(47 128)(48 129)(49 130)(50 131)(51 132)(52 133)(53 115)(54 116)(55 117)(56 118)(57 119)(58 88)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)(65 95)(66 77)(67 78)(68 79)(69 80)(70 81)(71 82)(72 83)(73 84)(74 85)(75 86)(76 87)
G:=sub<Sym(133)| (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), (1,49,66,113,77,130,23)(2,50,67,114,78,131,24)(3,51,68,96,79,132,25)(4,52,69,97,80,133,26)(5,53,70,98,81,115,27)(6,54,71,99,82,116,28)(7,55,72,100,83,117,29)(8,56,73,101,84,118,30)(9,57,74,102,85,119,31)(10,39,75,103,86,120,32)(11,40,76,104,87,121,33)(12,41,58,105,88,122,34)(13,42,59,106,89,123,35)(14,43,60,107,90,124,36)(15,44,61,108,91,125,37)(16,45,62,109,92,126,38)(17,46,63,110,93,127,20)(18,47,64,111,94,128,21)(19,48,65,112,95,129,22), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,20)(18,21)(19,22)(39,120)(40,121)(41,122)(42,123)(43,124)(44,125)(45,126)(46,127)(47,128)(48,129)(49,130)(50,131)(51,132)(52,133)(53,115)(54,116)(55,117)(56,118)(57,119)(58,88)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(65,95)(66,77)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87)>;
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), (1,49,66,113,77,130,23)(2,50,67,114,78,131,24)(3,51,68,96,79,132,25)(4,52,69,97,80,133,26)(5,53,70,98,81,115,27)(6,54,71,99,82,116,28)(7,55,72,100,83,117,29)(8,56,73,101,84,118,30)(9,57,74,102,85,119,31)(10,39,75,103,86,120,32)(11,40,76,104,87,121,33)(12,41,58,105,88,122,34)(13,42,59,106,89,123,35)(14,43,60,107,90,124,36)(15,44,61,108,91,125,37)(16,45,62,109,92,126,38)(17,46,63,110,93,127,20)(18,47,64,111,94,128,21)(19,48,65,112,95,129,22), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,20)(18,21)(19,22)(39,120)(40,121)(41,122)(42,123)(43,124)(44,125)(45,126)(46,127)(47,128)(48,129)(49,130)(50,131)(51,132)(52,133)(53,115)(54,116)(55,117)(56,118)(57,119)(58,88)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(65,95)(66,77)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87) );
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)], [(1,49,66,113,77,130,23),(2,50,67,114,78,131,24),(3,51,68,96,79,132,25),(4,52,69,97,80,133,26),(5,53,70,98,81,115,27),(6,54,71,99,82,116,28),(7,55,72,100,83,117,29),(8,56,73,101,84,118,30),(9,57,74,102,85,119,31),(10,39,75,103,86,120,32),(11,40,76,104,87,121,33),(12,41,58,105,88,122,34),(13,42,59,106,89,123,35),(14,43,60,107,90,124,36),(15,44,61,108,91,125,37),(16,45,62,109,92,126,38),(17,46,63,110,93,127,20),(18,47,64,111,94,128,21),(19,48,65,112,95,129,22)], [(1,23),(2,24),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,31),(10,32),(11,33),(12,34),(13,35),(14,36),(15,37),(16,38),(17,20),(18,21),(19,22),(39,120),(40,121),(41,122),(42,123),(43,124),(44,125),(45,126),(46,127),(47,128),(48,129),(49,130),(50,131),(51,132),(52,133),(53,115),(54,116),(55,117),(56,118),(57,119),(58,88),(59,89),(60,90),(61,91),(62,92),(63,93),(64,94),(65,95),(66,77),(67,78),(68,79),(69,80),(70,81),(71,82),(72,83),(73,84),(74,85),(75,86),(76,87)]])
95 conjugacy classes
| class | 1 | 2 | 7A | 7B | 7C | 19A | ··· | 19R | 38A | ··· | 38R | 133A | ··· | 133BB |
| order | 1 | 2 | 7 | 7 | 7 | 19 | ··· | 19 | 38 | ··· | 38 | 133 | ··· | 133 |
| size | 1 | 7 | 2 | 2 | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 |
95 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C19 | C38 | D7 | D7×C19 |
| kernel | D7×C19 | C133 | D7 | C7 | C19 | C1 |
| # reps | 1 | 1 | 18 | 18 | 3 | 54 |
Matrix representation of D7×C19 ►in GL2(𝔽1597) generated by
| 590 | 0 |
| 0 | 590 |
| 0 | 1 |
| 1596 | 1321 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(1597))| [590,0,0,590],[0,1596,1,1321],[0,1,1,0] >;
D7×C19 in GAP, Magma, Sage, TeX
D_7\times C_{19} % in TeX
G:=Group("D7xC19"); // GroupNames label
G:=SmallGroup(266,1);
// by ID
G=gap.SmallGroup(266,1);
# by ID
G:=PCGroup([3,-2,-19,-7,2054]);
// Polycyclic
G:=Group<a,b,c|a^19=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export