direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C19×C7⋊C3, C7⋊C57, C133⋊1C3, SmallGroup(399,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C19×C7⋊C3 |
Generators and relations for C19×C7⋊C3
G = < a,b,c | a19=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >
Smallest permutation representation of C19×C7⋊C3
►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 93 63 97 124 56 37)(2 94 64 98 125 57 38)(3 95 65 99 126 39 20)(4 77 66 100 127 40 21)(5 78 67 101 128 41 22)(6 79 68 102 129 42 23)(7 80 69 103 130 43 24)(8 81 70 104 131 44 25)(9 82 71 105 132 45 26)(10 83 72 106 133 46 27)(11 84 73 107 115 47 28)(12 85 74 108 116 48 29)(13 86 75 109 117 49 30)(14 87 76 110 118 50 31)(15 88 58 111 119 51 32)(16 89 59 112 120 52 33)(17 90 60 113 121 53 34)(18 91 61 114 122 54 35)(19 92 62 96 123 55 36)
(20 39 99)(21 40 100)(22 41 101)(23 42 102)(24 43 103)(25 44 104)(26 45 105)(27 46 106)(28 47 107)(29 48 108)(30 49 109)(31 50 110)(32 51 111)(33 52 112)(34 53 113)(35 54 114)(36 55 96)(37 56 97)(38 57 98)(58 119 88)(59 120 89)(60 121 90)(61 122 91)(62 123 92)(63 124 93)(64 125 94)(65 126 95)(66 127 77)(67 128 78)(68 129 79)(69 130 80)(70 131 81)(71 132 82)(72 133 83)(73 115 84)(74 116 85)(75 117 86)(76 118 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), (1,93,63,97,124,56,37)(2,94,64,98,125,57,38)(3,95,65,99,126,39,20)(4,77,66,100,127,40,21)(5,78,67,101,128,41,22)(6,79,68,102,129,42,23)(7,80,69,103,130,43,24)(8,81,70,104,131,44,25)(9,82,71,105,132,45,26)(10,83,72,106,133,46,27)(11,84,73,107,115,47,28)(12,85,74,108,116,48,29)(13,86,75,109,117,49,30)(14,87,76,110,118,50,31)(15,88,58,111,119,51,32)(16,89,59,112,120,52,33)(17,90,60,113,121,53,34)(18,91,61,114,122,54,35)(19,92,62,96,123,55,36), (20,39,99)(21,40,100)(22,41,101)(23,42,102)(24,43,103)(25,44,104)(26,45,105)(27,46,106)(28,47,107)(29,48,108)(30,49,109)(31,50,110)(32,51,111)(33,52,112)(34,53,113)(35,54,114)(36,55,96)(37,56,97)(38,57,98)(58,119,88)(59,120,89)(60,121,90)(61,122,91)(62,123,92)(63,124,93)(64,125,94)(65,126,95)(66,127,77)(67,128,78)(68,129,79)(69,130,80)(70,131,81)(71,132,82)(72,133,83)(73,115,84)(74,116,85)(75,117,86)(76,118,87)>;
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), (1,93,63,97,124,56,37)(2,94,64,98,125,57,38)(3,95,65,99,126,39,20)(4,77,66,100,127,40,21)(5,78,67,101,128,41,22)(6,79,68,102,129,42,23)(7,80,69,103,130,43,24)(8,81,70,104,131,44,25)(9,82,71,105,132,45,26)(10,83,72,106,133,46,27)(11,84,73,107,115,47,28)(12,85,74,108,116,48,29)(13,86,75,109,117,49,30)(14,87,76,110,118,50,31)(15,88,58,111,119,51,32)(16,89,59,112,120,52,33)(17,90,60,113,121,53,34)(18,91,61,114,122,54,35)(19,92,62,96,123,55,36), (20,39,99)(21,40,100)(22,41,101)(23,42,102)(24,43,103)(25,44,104)(26,45,105)(27,46,106)(28,47,107)(29,48,108)(30,49,109)(31,50,110)(32,51,111)(33,52,112)(34,53,113)(35,54,114)(36,55,96)(37,56,97)(38,57,98)(58,119,88)(59,120,89)(60,121,90)(61,122,91)(62,123,92)(63,124,93)(64,125,94)(65,126,95)(66,127,77)(67,128,78)(68,129,79)(69,130,80)(70,131,81)(71,132,82)(72,133,83)(73,115,84)(74,116,85)(75,117,86)(76,118,87) );
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)], [(1,93,63,97,124,56,37),(2,94,64,98,125,57,38),(3,95,65,99,126,39,20),(4,77,66,100,127,40,21),(5,78,67,101,128,41,22),(6,79,68,102,129,42,23),(7,80,69,103,130,43,24),(8,81,70,104,131,44,25),(9,82,71,105,132,45,26),(10,83,72,106,133,46,27),(11,84,73,107,115,47,28),(12,85,74,108,116,48,29),(13,86,75,109,117,49,30),(14,87,76,110,118,50,31),(15,88,58,111,119,51,32),(16,89,59,112,120,52,33),(17,90,60,113,121,53,34),(18,91,61,114,122,54,35),(19,92,62,96,123,55,36)], [(20,39,99),(21,40,100),(22,41,101),(23,42,102),(24,43,103),(25,44,104),(26,45,105),(27,46,106),(28,47,107),(29,48,108),(30,49,109),(31,50,110),(32,51,111),(33,52,112),(34,53,113),(35,54,114),(36,55,96),(37,56,97),(38,57,98),(58,119,88),(59,120,89),(60,121,90),(61,122,91),(62,123,92),(63,124,93),(64,125,94),(65,126,95),(66,127,77),(67,128,78),(68,129,79),(69,130,80),(70,131,81),(71,132,82),(72,133,83),(73,115,84),(74,116,85),(75,117,86),(76,118,87)]])
95 conjugacy classes
| class | 1 | 3A | 3B | 7A | 7B | 19A | ··· | 19R | 57A | ··· | 57AJ | 133A | ··· | 133AJ |
| order | 1 | 3 | 3 | 7 | 7 | 19 | ··· | 19 | 57 | ··· | 57 | 133 | ··· | 133 |
| size | 1 | 7 | 7 | 3 | 3 | 1 | ··· | 1 | 7 | ··· | 7 | 3 | ··· | 3 |
95 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | |||||
| image | C1 | C3 | C19 | C57 | C7⋊C3 | C19×C7⋊C3 |
| kernel | C19×C7⋊C3 | C133 | C7⋊C3 | C7 | C19 | C1 |
| # reps | 1 | 2 | 18 | 36 | 2 | 36 |
Matrix representation of C19×C7⋊C3 ►in GL3(𝔽1597) generated by
| 81 | 0 | 0 |
| 0 | 81 | 0 |
| 0 | 0 | 81 |
| 57 | 1 | 1540 |
| 1 | 0 | 1596 |
| 0 | 1 | 1596 |
| 1596 | 0 | 1 |
| 56 | 1 | 1541 |
| 1596 | 0 | 0 |
G:=sub<GL(3,GF(1597))| [81,0,0,0,81,0,0,0,81],[57,1,0,1,0,1,1540,1596,1596],[1596,56,1596,0,1,0,1,1541,0] >;
C19×C7⋊C3 in GAP, Magma, Sage, TeX
C_{19}\times C_7\rtimes C_3 % in TeX
G:=Group("C19xC7:C3"); // GroupNames label
G:=SmallGroup(399,1);
// by ID
G=gap.SmallGroup(399,1);
# by ID
G:=PCGroup([3,-3,-19,-7,1028]);
// Polycyclic
G:=Group<a,b,c|a^19=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
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