metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊3D19, D38.C4, C152⋊4C2, C4.13D38, Dic19.C4, C19⋊1M4(2), C76.13C22, C19⋊C8⋊4C2, C38.2(C2×C4), C2.3(C4×D19), (C4×D19).2C2, SmallGroup(304,4)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊D19
G = < a,b,c | a8=b19=c2=1, ab=ba, cac=a5, cbc=b-1 >
Smallest permutation representation of C8⋊D19
►On 152 points
Generators in S152
(1 143 63 110 36 125 54 86)(2 144 64 111 37 126 55 87)(3 145 65 112 38 127 56 88)(4 146 66 113 20 128 57 89)(5 147 67 114 21 129 39 90)(6 148 68 96 22 130 40 91)(7 149 69 97 23 131 41 92)(8 150 70 98 24 132 42 93)(9 151 71 99 25 133 43 94)(10 152 72 100 26 115 44 95)(11 134 73 101 27 116 45 77)(12 135 74 102 28 117 46 78)(13 136 75 103 29 118 47 79)(14 137 76 104 30 119 48 80)(15 138 58 105 31 120 49 81)(16 139 59 106 32 121 50 82)(17 140 60 107 33 122 51 83)(18 141 61 108 34 123 52 84)(19 142 62 109 35 124 53 85)
(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 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 32)(21 31)(22 30)(23 29)(24 28)(25 27)(33 38)(34 37)(35 36)(39 49)(40 48)(41 47)(42 46)(43 45)(50 57)(51 56)(52 55)(53 54)(58 67)(59 66)(60 65)(61 64)(62 63)(68 76)(69 75)(70 74)(71 73)(77 99)(78 98)(79 97)(80 96)(81 114)(82 113)(83 112)(84 111)(85 110)(86 109)(87 108)(88 107)(89 106)(90 105)(91 104)(92 103)(93 102)(94 101)(95 100)(115 152)(116 151)(117 150)(118 149)(119 148)(120 147)(121 146)(122 145)(123 144)(124 143)(125 142)(126 141)(127 140)(128 139)(129 138)(130 137)(131 136)(132 135)(133 134)
G:=sub<Sym(152)| (1,143,63,110,36,125,54,86)(2,144,64,111,37,126,55,87)(3,145,65,112,38,127,56,88)(4,146,66,113,20,128,57,89)(5,147,67,114,21,129,39,90)(6,148,68,96,22,130,40,91)(7,149,69,97,23,131,41,92)(8,150,70,98,24,132,42,93)(9,151,71,99,25,133,43,94)(10,152,72,100,26,115,44,95)(11,134,73,101,27,116,45,77)(12,135,74,102,28,117,46,78)(13,136,75,103,29,118,47,79)(14,137,76,104,30,119,48,80)(15,138,58,105,31,120,49,81)(16,139,59,106,32,121,50,82)(17,140,60,107,33,122,51,83)(18,141,61,108,34,123,52,84)(19,142,62,109,35,124,53,85), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,32)(21,31)(22,30)(23,29)(24,28)(25,27)(33,38)(34,37)(35,36)(39,49)(40,48)(41,47)(42,46)(43,45)(50,57)(51,56)(52,55)(53,54)(58,67)(59,66)(60,65)(61,64)(62,63)(68,76)(69,75)(70,74)(71,73)(77,99)(78,98)(79,97)(80,96)(81,114)(82,113)(83,112)(84,111)(85,110)(86,109)(87,108)(88,107)(89,106)(90,105)(91,104)(92,103)(93,102)(94,101)(95,100)(115,152)(116,151)(117,150)(118,149)(119,148)(120,147)(121,146)(122,145)(123,144)(124,143)(125,142)(126,141)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134)>;
G:=Group( (1,143,63,110,36,125,54,86)(2,144,64,111,37,126,55,87)(3,145,65,112,38,127,56,88)(4,146,66,113,20,128,57,89)(5,147,67,114,21,129,39,90)(6,148,68,96,22,130,40,91)(7,149,69,97,23,131,41,92)(8,150,70,98,24,132,42,93)(9,151,71,99,25,133,43,94)(10,152,72,100,26,115,44,95)(11,134,73,101,27,116,45,77)(12,135,74,102,28,117,46,78)(13,136,75,103,29,118,47,79)(14,137,76,104,30,119,48,80)(15,138,58,105,31,120,49,81)(16,139,59,106,32,121,50,82)(17,140,60,107,33,122,51,83)(18,141,61,108,34,123,52,84)(19,142,62,109,35,124,53,85), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,32)(21,31)(22,30)(23,29)(24,28)(25,27)(33,38)(34,37)(35,36)(39,49)(40,48)(41,47)(42,46)(43,45)(50,57)(51,56)(52,55)(53,54)(58,67)(59,66)(60,65)(61,64)(62,63)(68,76)(69,75)(70,74)(71,73)(77,99)(78,98)(79,97)(80,96)(81,114)(82,113)(83,112)(84,111)(85,110)(86,109)(87,108)(88,107)(89,106)(90,105)(91,104)(92,103)(93,102)(94,101)(95,100)(115,152)(116,151)(117,150)(118,149)(119,148)(120,147)(121,146)(122,145)(123,144)(124,143)(125,142)(126,141)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134) );
G=PermutationGroup([[(1,143,63,110,36,125,54,86),(2,144,64,111,37,126,55,87),(3,145,65,112,38,127,56,88),(4,146,66,113,20,128,57,89),(5,147,67,114,21,129,39,90),(6,148,68,96,22,130,40,91),(7,149,69,97,23,131,41,92),(8,150,70,98,24,132,42,93),(9,151,71,99,25,133,43,94),(10,152,72,100,26,115,44,95),(11,134,73,101,27,116,45,77),(12,135,74,102,28,117,46,78),(13,136,75,103,29,118,47,79),(14,137,76,104,30,119,48,80),(15,138,58,105,31,120,49,81),(16,139,59,106,32,121,50,82),(17,140,60,107,33,122,51,83),(18,141,61,108,34,123,52,84),(19,142,62,109,35,124,53,85)], [(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)], [(1,19),(2,18),(3,17),(4,16),(5,15),(6,14),(7,13),(8,12),(9,11),(20,32),(21,31),(22,30),(23,29),(24,28),(25,27),(33,38),(34,37),(35,36),(39,49),(40,48),(41,47),(42,46),(43,45),(50,57),(51,56),(52,55),(53,54),(58,67),(59,66),(60,65),(61,64),(62,63),(68,76),(69,75),(70,74),(71,73),(77,99),(78,98),(79,97),(80,96),(81,114),(82,113),(83,112),(84,111),(85,110),(86,109),(87,108),(88,107),(89,106),(90,105),(91,104),(92,103),(93,102),(94,101),(95,100),(115,152),(116,151),(117,150),(118,149),(119,148),(120,147),(121,146),(122,145),(123,144),(124,143),(125,142),(126,141),(127,140),(128,139),(129,138),(130,137),(131,136),(132,135),(133,134)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 19A | ··· | 19I | 38A | ··· | 38I | 76A | ··· | 76R | 152A | ··· | 152AJ |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 | 152 | ··· | 152 |
| size | 1 | 1 | 38 | 1 | 1 | 38 | 2 | 2 | 38 | 38 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | M4(2) | D19 | D38 | C4×D19 | C8⋊D19 |
| kernel | C8⋊D19 | C19⋊C8 | C152 | C4×D19 | Dic19 | D38 | C19 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 9 | 9 | 18 | 36 |
Matrix representation of C8⋊D19 ►in GL2(𝔽37) generated by
| 10 | 28 |
| 20 | 27 |
| 7 | 29 |
| 26 | 18 |
| 19 | 26 |
| 26 | 18 |
G:=sub<GL(2,GF(37))| [10,20,28,27],[7,26,29,18],[19,26,26,18] >;
C8⋊D19 in GAP, Magma, Sage, TeX
C_8\rtimes D_{19} % in TeX
G:=Group("C8:D19"); // GroupNames label
G:=SmallGroup(304,4);
// by ID
G=gap.SmallGroup(304,4);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-19,101,26,42,7204]);
// Polycyclic
G:=Group<a,b,c|a^8=b^19=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations
Export