direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C8×D19, C152⋊3C2, D38.2C4, C4.12D38, C76.12C22, Dic19.2C4, C19⋊C8⋊6C2, C19⋊1(C2×C8), C38.1(C2×C4), C2.1(C4×D19), (C4×D19).3C2, SmallGroup(304,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C8×D19 |
Generators and relations for C8×D19
G = < a,b,c | a8=b19=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C8×D19
►On 152 points
Generators in S152
(1 144 59 101 28 132 45 81)(2 145 60 102 29 133 46 82)(3 146 61 103 30 115 47 83)(4 147 62 104 31 116 48 84)(5 148 63 105 32 117 49 85)(6 149 64 106 33 118 50 86)(7 150 65 107 34 119 51 87)(8 151 66 108 35 120 52 88)(9 152 67 109 36 121 53 89)(10 134 68 110 37 122 54 90)(11 135 69 111 38 123 55 91)(12 136 70 112 20 124 56 92)(13 137 71 113 21 125 57 93)(14 138 72 114 22 126 39 94)(15 139 73 96 23 127 40 95)(16 140 74 97 24 128 41 77)(17 141 75 98 25 129 42 78)(18 142 76 99 26 130 43 79)(19 143 58 100 27 131 44 80)
(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 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 20)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(39 64)(40 63)(41 62)(42 61)(43 60)(44 59)(45 58)(46 76)(47 75)(48 74)(49 73)(50 72)(51 71)(52 70)(53 69)(54 68)(55 67)(56 66)(57 65)(77 104)(78 103)(79 102)(80 101)(81 100)(82 99)(83 98)(84 97)(85 96)(86 114)(87 113)(88 112)(89 111)(90 110)(91 109)(92 108)(93 107)(94 106)(95 105)(115 141)(116 140)(117 139)(118 138)(119 137)(120 136)(121 135)(122 134)(123 152)(124 151)(125 150)(126 149)(127 148)(128 147)(129 146)(130 145)(131 144)(132 143)(133 142)
G:=sub<Sym(152)| (1,144,59,101,28,132,45,81)(2,145,60,102,29,133,46,82)(3,146,61,103,30,115,47,83)(4,147,62,104,31,116,48,84)(5,148,63,105,32,117,49,85)(6,149,64,106,33,118,50,86)(7,150,65,107,34,119,51,87)(8,151,66,108,35,120,52,88)(9,152,67,109,36,121,53,89)(10,134,68,110,37,122,54,90)(11,135,69,111,38,123,55,91)(12,136,70,112,20,124,56,92)(13,137,71,113,21,125,57,93)(14,138,72,114,22,126,39,94)(15,139,73,96,23,127,40,95)(16,140,74,97,24,128,41,77)(17,141,75,98,25,129,42,78)(18,142,76,99,26,130,43,79)(19,143,58,100,27,131,44,80), (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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,76)(47,75)(48,74)(49,73)(50,72)(51,71)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(77,104)(78,103)(79,102)(80,101)(81,100)(82,99)(83,98)(84,97)(85,96)(86,114)(87,113)(88,112)(89,111)(90,110)(91,109)(92,108)(93,107)(94,106)(95,105)(115,141)(116,140)(117,139)(118,138)(119,137)(120,136)(121,135)(122,134)(123,152)(124,151)(125,150)(126,149)(127,148)(128,147)(129,146)(130,145)(131,144)(132,143)(133,142)>;
G:=Group( (1,144,59,101,28,132,45,81)(2,145,60,102,29,133,46,82)(3,146,61,103,30,115,47,83)(4,147,62,104,31,116,48,84)(5,148,63,105,32,117,49,85)(6,149,64,106,33,118,50,86)(7,150,65,107,34,119,51,87)(8,151,66,108,35,120,52,88)(9,152,67,109,36,121,53,89)(10,134,68,110,37,122,54,90)(11,135,69,111,38,123,55,91)(12,136,70,112,20,124,56,92)(13,137,71,113,21,125,57,93)(14,138,72,114,22,126,39,94)(15,139,73,96,23,127,40,95)(16,140,74,97,24,128,41,77)(17,141,75,98,25,129,42,78)(18,142,76,99,26,130,43,79)(19,143,58,100,27,131,44,80), (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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,76)(47,75)(48,74)(49,73)(50,72)(51,71)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(77,104)(78,103)(79,102)(80,101)(81,100)(82,99)(83,98)(84,97)(85,96)(86,114)(87,113)(88,112)(89,111)(90,110)(91,109)(92,108)(93,107)(94,106)(95,105)(115,141)(116,140)(117,139)(118,138)(119,137)(120,136)(121,135)(122,134)(123,152)(124,151)(125,150)(126,149)(127,148)(128,147)(129,146)(130,145)(131,144)(132,143)(133,142) );
G=PermutationGroup([[(1,144,59,101,28,132,45,81),(2,145,60,102,29,133,46,82),(3,146,61,103,30,115,47,83),(4,147,62,104,31,116,48,84),(5,148,63,105,32,117,49,85),(6,149,64,106,33,118,50,86),(7,150,65,107,34,119,51,87),(8,151,66,108,35,120,52,88),(9,152,67,109,36,121,53,89),(10,134,68,110,37,122,54,90),(11,135,69,111,38,123,55,91),(12,136,70,112,20,124,56,92),(13,137,71,113,21,125,57,93),(14,138,72,114,22,126,39,94),(15,139,73,96,23,127,40,95),(16,140,74,97,24,128,41,77),(17,141,75,98,25,129,42,78),(18,142,76,99,26,130,43,79),(19,143,58,100,27,131,44,80)], [(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)], [(1,27),(2,26),(3,25),(4,24),(5,23),(6,22),(7,21),(8,20),(9,38),(10,37),(11,36),(12,35),(13,34),(14,33),(15,32),(16,31),(17,30),(18,29),(19,28),(39,64),(40,63),(41,62),(42,61),(43,60),(44,59),(45,58),(46,76),(47,75),(48,74),(49,73),(50,72),(51,71),(52,70),(53,69),(54,68),(55,67),(56,66),(57,65),(77,104),(78,103),(79,102),(80,101),(81,100),(82,99),(83,98),(84,97),(85,96),(86,114),(87,113),(88,112),(89,111),(90,110),(91,109),(92,108),(93,107),(94,106),(95,105),(115,141),(116,140),(117,139),(118,138),(119,137),(120,136),(121,135),(122,134),(123,152),(124,151),(125,150),(126,149),(127,148),(128,147),(129,146),(130,145),(131,144),(132,143),(133,142)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 19A | ··· | 19I | 38A | ··· | 38I | 76A | ··· | 76R | 152A | ··· | 152AJ |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 | 152 | ··· | 152 |
| size | 1 | 1 | 19 | 19 | 1 | 1 | 19 | 19 | 1 | 1 | 1 | 1 | 19 | 19 | 19 | 19 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D19 | D38 | C4×D19 | C8×D19 |
| kernel | C8×D19 | C19⋊C8 | C152 | C4×D19 | Dic19 | D38 | D19 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 9 | 9 | 18 | 36 |
Matrix representation of C8×D19 ►in GL2(𝔽457) generated by
| 170 | 0 |
| 0 | 170 |
| 100 | 1 |
| 394 | 301 |
| 156 | 1 |
| 343 | 301 |
G:=sub<GL(2,GF(457))| [170,0,0,170],[100,394,1,301],[156,343,1,301] >;
C8×D19 in GAP, Magma, Sage, TeX
C_8\times D_{19} % in TeX
G:=Group("C8xD19"); // GroupNames label
G:=SmallGroup(304,3);
// by ID
G=gap.SmallGroup(304,3);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-19,26,42,7204]);
// Polycyclic
G:=Group<a,b,c|a^8=b^19=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export