direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C7×C19⋊C3, C19⋊C21, C133⋊2C3, SmallGroup(399,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C7×C19⋊C3 |
Generators and relations for C7×C19⋊C3
G = < a,b,c | a7=b19=c3=1, ab=ba, ac=ca, cbc-1=b11 >
Smallest permutation representation of C7×C19⋊C3
►On 133 points
Generators in S133
(1 115 96 77 58 39 20)(2 116 97 78 59 40 21)(3 117 98 79 60 41 22)(4 118 99 80 61 42 23)(5 119 100 81 62 43 24)(6 120 101 82 63 44 25)(7 121 102 83 64 45 26)(8 122 103 84 65 46 27)(9 123 104 85 66 47 28)(10 124 105 86 67 48 29)(11 125 106 87 68 49 30)(12 126 107 88 69 50 31)(13 127 108 89 70 51 32)(14 128 109 90 71 52 33)(15 129 110 91 72 53 34)(16 130 111 92 73 54 35)(17 131 112 93 74 55 36)(18 132 113 94 75 56 37)(19 133 114 95 76 57 38)
(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)
(2 8 12)(3 15 4)(5 10 7)(6 17 18)(9 19 13)(11 14 16)(21 27 31)(22 34 23)(24 29 26)(25 36 37)(28 38 32)(30 33 35)(40 46 50)(41 53 42)(43 48 45)(44 55 56)(47 57 51)(49 52 54)(59 65 69)(60 72 61)(62 67 64)(63 74 75)(66 76 70)(68 71 73)(78 84 88)(79 91 80)(81 86 83)(82 93 94)(85 95 89)(87 90 92)(97 103 107)(98 110 99)(100 105 102)(101 112 113)(104 114 108)(106 109 111)(116 122 126)(117 129 118)(119 124 121)(120 131 132)(123 133 127)(125 128 130)
G:=sub<Sym(133)| (1,115,96,77,58,39,20)(2,116,97,78,59,40,21)(3,117,98,79,60,41,22)(4,118,99,80,61,42,23)(5,119,100,81,62,43,24)(6,120,101,82,63,44,25)(7,121,102,83,64,45,26)(8,122,103,84,65,46,27)(9,123,104,85,66,47,28)(10,124,105,86,67,48,29)(11,125,106,87,68,49,30)(12,126,107,88,69,50,31)(13,127,108,89,70,51,32)(14,128,109,90,71,52,33)(15,129,110,91,72,53,34)(16,130,111,92,73,54,35)(17,131,112,93,74,55,36)(18,132,113,94,75,56,37)(19,133,114,95,76,57,38), (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), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16)(21,27,31)(22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)(40,46,50)(41,53,42)(43,48,45)(44,55,56)(47,57,51)(49,52,54)(59,65,69)(60,72,61)(62,67,64)(63,74,75)(66,76,70)(68,71,73)(78,84,88)(79,91,80)(81,86,83)(82,93,94)(85,95,89)(87,90,92)(97,103,107)(98,110,99)(100,105,102)(101,112,113)(104,114,108)(106,109,111)(116,122,126)(117,129,118)(119,124,121)(120,131,132)(123,133,127)(125,128,130)>;
G:=Group( (1,115,96,77,58,39,20)(2,116,97,78,59,40,21)(3,117,98,79,60,41,22)(4,118,99,80,61,42,23)(5,119,100,81,62,43,24)(6,120,101,82,63,44,25)(7,121,102,83,64,45,26)(8,122,103,84,65,46,27)(9,123,104,85,66,47,28)(10,124,105,86,67,48,29)(11,125,106,87,68,49,30)(12,126,107,88,69,50,31)(13,127,108,89,70,51,32)(14,128,109,90,71,52,33)(15,129,110,91,72,53,34)(16,130,111,92,73,54,35)(17,131,112,93,74,55,36)(18,132,113,94,75,56,37)(19,133,114,95,76,57,38), (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), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16)(21,27,31)(22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)(40,46,50)(41,53,42)(43,48,45)(44,55,56)(47,57,51)(49,52,54)(59,65,69)(60,72,61)(62,67,64)(63,74,75)(66,76,70)(68,71,73)(78,84,88)(79,91,80)(81,86,83)(82,93,94)(85,95,89)(87,90,92)(97,103,107)(98,110,99)(100,105,102)(101,112,113)(104,114,108)(106,109,111)(116,122,126)(117,129,118)(119,124,121)(120,131,132)(123,133,127)(125,128,130) );
G=PermutationGroup([[(1,115,96,77,58,39,20),(2,116,97,78,59,40,21),(3,117,98,79,60,41,22),(4,118,99,80,61,42,23),(5,119,100,81,62,43,24),(6,120,101,82,63,44,25),(7,121,102,83,64,45,26),(8,122,103,84,65,46,27),(9,123,104,85,66,47,28),(10,124,105,86,67,48,29),(11,125,106,87,68,49,30),(12,126,107,88,69,50,31),(13,127,108,89,70,51,32),(14,128,109,90,71,52,33),(15,129,110,91,72,53,34),(16,130,111,92,73,54,35),(17,131,112,93,74,55,36),(18,132,113,94,75,56,37),(19,133,114,95,76,57,38)], [(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)], [(2,8,12),(3,15,4),(5,10,7),(6,17,18),(9,19,13),(11,14,16),(21,27,31),(22,34,23),(24,29,26),(25,36,37),(28,38,32),(30,33,35),(40,46,50),(41,53,42),(43,48,45),(44,55,56),(47,57,51),(49,52,54),(59,65,69),(60,72,61),(62,67,64),(63,74,75),(66,76,70),(68,71,73),(78,84,88),(79,91,80),(81,86,83),(82,93,94),(85,95,89),(87,90,92),(97,103,107),(98,110,99),(100,105,102),(101,112,113),(104,114,108),(106,109,111),(116,122,126),(117,129,118),(119,124,121),(120,131,132),(123,133,127),(125,128,130)]])
63 conjugacy classes
| class | 1 | 3A | 3B | 7A | ··· | 7F | 19A | ··· | 19F | 21A | ··· | 21L | 133A | ··· | 133AJ |
| order | 1 | 3 | 3 | 7 | ··· | 7 | 19 | ··· | 19 | 21 | ··· | 21 | 133 | ··· | 133 |
| size | 1 | 19 | 19 | 1 | ··· | 1 | 3 | ··· | 3 | 19 | ··· | 19 | 3 | ··· | 3 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | |||||
| image | C1 | C3 | C7 | C21 | C19⋊C3 | C7×C19⋊C3 |
| kernel | C7×C19⋊C3 | C133 | C19⋊C3 | C19 | C7 | C1 |
| # reps | 1 | 2 | 6 | 12 | 6 | 36 |
Matrix representation of C7×C19⋊C3 ►in GL3(𝔽1597) generated by
| 319 | 0 | 0 |
| 0 | 319 | 0 |
| 0 | 0 | 319 |
| 473 | 719 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1352 | 718 | 11 |
| 1 | 1124 | 878 |
G:=sub<GL(3,GF(1597))| [319,0,0,0,319,0,0,0,319],[473,1,0,719,0,1,1,0,0],[1,1352,1,0,718,1124,0,11,878] >;
C7×C19⋊C3 in GAP, Magma, Sage, TeX
C_7\times C_{19}\rtimes C_3 % in TeX
G:=Group("C7xC19:C3"); // GroupNames label
G:=SmallGroup(399,2);
// by ID
G=gap.SmallGroup(399,2);
# by ID
G:=PCGroup([3,-3,-7,-19,1325]);
// Polycyclic
G:=Group<a,b,c|a^7=b^19=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^11>;
// generators/relations
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