metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C19⋊C8, C38.C4, C76.2C2, C4.2D19, C2.Dic19, SmallGroup(152,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C19⋊C8 |
Generators and relations for C19⋊C8
G = < a,b | a19=b8=1, bab-1=a-1 >
Smallest permutation representation of C19⋊C8
►Regular action on 152 points
Generators in S152
(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)(134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152)
(1 134 76 96 34 115 54 77)(2 152 58 114 35 133 55 95)(3 151 59 113 36 132 56 94)(4 150 60 112 37 131 57 93)(5 149 61 111 38 130 39 92)(6 148 62 110 20 129 40 91)(7 147 63 109 21 128 41 90)(8 146 64 108 22 127 42 89)(9 145 65 107 23 126 43 88)(10 144 66 106 24 125 44 87)(11 143 67 105 25 124 45 86)(12 142 68 104 26 123 46 85)(13 141 69 103 27 122 47 84)(14 140 70 102 28 121 48 83)(15 139 71 101 29 120 49 82)(16 138 72 100 30 119 50 81)(17 137 73 99 31 118 51 80)(18 136 74 98 32 117 52 79)(19 135 75 97 33 116 53 78)
G:=sub<Sym(152)| (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)(134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152), (1,134,76,96,34,115,54,77)(2,152,58,114,35,133,55,95)(3,151,59,113,36,132,56,94)(4,150,60,112,37,131,57,93)(5,149,61,111,38,130,39,92)(6,148,62,110,20,129,40,91)(7,147,63,109,21,128,41,90)(8,146,64,108,22,127,42,89)(9,145,65,107,23,126,43,88)(10,144,66,106,24,125,44,87)(11,143,67,105,25,124,45,86)(12,142,68,104,26,123,46,85)(13,141,69,103,27,122,47,84)(14,140,70,102,28,121,48,83)(15,139,71,101,29,120,49,82)(16,138,72,100,30,119,50,81)(17,137,73,99,31,118,51,80)(18,136,74,98,32,117,52,79)(19,135,75,97,33,116,53,78)>;
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)(134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152), (1,134,76,96,34,115,54,77)(2,152,58,114,35,133,55,95)(3,151,59,113,36,132,56,94)(4,150,60,112,37,131,57,93)(5,149,61,111,38,130,39,92)(6,148,62,110,20,129,40,91)(7,147,63,109,21,128,41,90)(8,146,64,108,22,127,42,89)(9,145,65,107,23,126,43,88)(10,144,66,106,24,125,44,87)(11,143,67,105,25,124,45,86)(12,142,68,104,26,123,46,85)(13,141,69,103,27,122,47,84)(14,140,70,102,28,121,48,83)(15,139,71,101,29,120,49,82)(16,138,72,100,30,119,50,81)(17,137,73,99,31,118,51,80)(18,136,74,98,32,117,52,79)(19,135,75,97,33,116,53,78) );
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),(134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152)], [(1,134,76,96,34,115,54,77),(2,152,58,114,35,133,55,95),(3,151,59,113,36,132,56,94),(4,150,60,112,37,131,57,93),(5,149,61,111,38,130,39,92),(6,148,62,110,20,129,40,91),(7,147,63,109,21,128,41,90),(8,146,64,108,22,127,42,89),(9,145,65,107,23,126,43,88),(10,144,66,106,24,125,44,87),(11,143,67,105,25,124,45,86),(12,142,68,104,26,123,46,85),(13,141,69,103,27,122,47,84),(14,140,70,102,28,121,48,83),(15,139,71,101,29,120,49,82),(16,138,72,100,30,119,50,81),(17,137,73,99,31,118,51,80),(18,136,74,98,32,117,52,79),(19,135,75,97,33,116,53,78)]])
C19⋊C8 is a maximal subgroup of
C8×D19 C8⋊D19 C76.C4 D4⋊D19 D4.D19 Q8⋊D19 C19⋊Q16 C19⋊C24 C57⋊C8
C19⋊C8 is a maximal quotient of
C19⋊C16 C57⋊C8
44 conjugacy classes
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 19A | ··· | 19I | 38A | ··· | 38I | 76A | ··· | 76R |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 1 | 1 | 19 | 19 | 19 | 19 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | - | |||
| image | C1 | C2 | C4 | C8 | D19 | Dic19 | C19⋊C8 |
| kernel | C19⋊C8 | C76 | C38 | C19 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 9 | 9 | 18 |
Matrix representation of C19⋊C8 ►in GL2(𝔽37) generated by
| 6 | 28 |
| 20 | 1 |
| 0 | 6 |
| 1 | 0 |
G:=sub<GL(2,GF(37))| [6,20,28,1],[0,1,6,0] >;
C19⋊C8 in GAP, Magma, Sage, TeX
C_{19}\rtimes C_8 % in TeX
G:=Group("C19:C8"); // GroupNames label
G:=SmallGroup(152,1);
// by ID
G=gap.SmallGroup(152,1);
# by ID
G:=PCGroup([4,-2,-2,-2,-19,8,21,2307]);
// Polycyclic
G:=Group<a,b|a^19=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
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