direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C8×C19⋊C3, C152⋊C3, C19⋊2C24, C76.4C6, C38.2C12, C2.(C4×C19⋊C3), C4.2(C2×C19⋊C3), (C4×C19⋊C3).4C2, (C2×C19⋊C3).2C4, SmallGroup(456,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C8×C19⋊C3 |
Generators and relations for C8×C19⋊C3
G = < a,b,c | a8=b19=c3=1, ab=ba, ac=ca, cbc-1=b11 >
Smallest permutation representation of C8×C19⋊C3
►On 152 points
Generators in S152
(1 134 58 96 20 115 39 77)(2 135 59 97 21 116 40 78)(3 136 60 98 22 117 41 79)(4 137 61 99 23 118 42 80)(5 138 62 100 24 119 43 81)(6 139 63 101 25 120 44 82)(7 140 64 102 26 121 45 83)(8 141 65 103 27 122 46 84)(9 142 66 104 28 123 47 85)(10 143 67 105 29 124 48 86)(11 144 68 106 30 125 49 87)(12 145 69 107 31 126 50 88)(13 146 70 108 32 127 51 89)(14 147 71 109 33 128 52 90)(15 148 72 110 34 129 53 91)(16 149 73 111 35 130 54 92)(17 150 74 112 36 131 55 93)(18 151 75 113 37 132 56 94)(19 152 76 114 38 133 57 95)
(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)
(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)(135 141 145)(136 148 137)(138 143 140)(139 150 151)(142 152 146)(144 147 149)
G:=sub<Sym(152)| (1,134,58,96,20,115,39,77)(2,135,59,97,21,116,40,78)(3,136,60,98,22,117,41,79)(4,137,61,99,23,118,42,80)(5,138,62,100,24,119,43,81)(6,139,63,101,25,120,44,82)(7,140,64,102,26,121,45,83)(8,141,65,103,27,122,46,84)(9,142,66,104,28,123,47,85)(10,143,67,105,29,124,48,86)(11,144,68,106,30,125,49,87)(12,145,69,107,31,126,50,88)(13,146,70,108,32,127,51,89)(14,147,71,109,33,128,52,90)(15,148,72,110,34,129,53,91)(16,149,73,111,35,130,54,92)(17,150,74,112,36,131,55,93)(18,151,75,113,37,132,56,94)(19,152,76,114,38,133,57,95), (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), (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)(135,141,145)(136,148,137)(138,143,140)(139,150,151)(142,152,146)(144,147,149)>;
G:=Group( (1,134,58,96,20,115,39,77)(2,135,59,97,21,116,40,78)(3,136,60,98,22,117,41,79)(4,137,61,99,23,118,42,80)(5,138,62,100,24,119,43,81)(6,139,63,101,25,120,44,82)(7,140,64,102,26,121,45,83)(8,141,65,103,27,122,46,84)(9,142,66,104,28,123,47,85)(10,143,67,105,29,124,48,86)(11,144,68,106,30,125,49,87)(12,145,69,107,31,126,50,88)(13,146,70,108,32,127,51,89)(14,147,71,109,33,128,52,90)(15,148,72,110,34,129,53,91)(16,149,73,111,35,130,54,92)(17,150,74,112,36,131,55,93)(18,151,75,113,37,132,56,94)(19,152,76,114,38,133,57,95), (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), (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)(135,141,145)(136,148,137)(138,143,140)(139,150,151)(142,152,146)(144,147,149) );
G=PermutationGroup([[(1,134,58,96,20,115,39,77),(2,135,59,97,21,116,40,78),(3,136,60,98,22,117,41,79),(4,137,61,99,23,118,42,80),(5,138,62,100,24,119,43,81),(6,139,63,101,25,120,44,82),(7,140,64,102,26,121,45,83),(8,141,65,103,27,122,46,84),(9,142,66,104,28,123,47,85),(10,143,67,105,29,124,48,86),(11,144,68,106,30,125,49,87),(12,145,69,107,31,126,50,88),(13,146,70,108,32,127,51,89),(14,147,71,109,33,128,52,90),(15,148,72,110,34,129,53,91),(16,149,73,111,35,130,54,92),(17,150,74,112,36,131,55,93),(18,151,75,113,37,132,56,94),(19,152,76,114,38,133,57,95)], [(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)], [(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),(135,141,145),(136,148,137),(138,143,140),(139,150,151),(142,152,146),(144,147,149)]])
72 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 19A | ··· | 19F | 24A | ··· | 24H | 38A | ··· | 38F | 76A | ··· | 76L | 152A | ··· | 152X |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 19 | ··· | 19 | 24 | ··· | 24 | 38 | ··· | 38 | 76 | ··· | 76 | 152 | ··· | 152 |
| size | 1 | 1 | 19 | 19 | 1 | 1 | 19 | 19 | 1 | 1 | 1 | 1 | 19 | 19 | 19 | 19 | 3 | ··· | 3 | 19 | ··· | 19 | 3 | ··· | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C24 | C19⋊C3 | C2×C19⋊C3 | C4×C19⋊C3 | C8×C19⋊C3 |
| kernel | C8×C19⋊C3 | C4×C19⋊C3 | C152 | C2×C19⋊C3 | C76 | C19⋊C3 | C38 | C19 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 6 | 6 | 12 | 24 |
Matrix representation of C8×C19⋊C3 ►in GL3(𝔽457) generated by
| 250 | 0 | 0 |
| 0 | 250 | 0 |
| 0 | 0 | 250 |
| 131 | 351 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 238 | 350 | 418 |
| 1 | 326 | 106 |
G:=sub<GL(3,GF(457))| [250,0,0,0,250,0,0,0,250],[131,1,0,351,0,1,1,0,0],[1,238,1,0,350,326,0,418,106] >;
C8×C19⋊C3 in GAP, Magma, Sage, TeX
C_8\times C_{19}\rtimes C_3 % in TeX
G:=Group("C8xC19:C3"); // GroupNames label
G:=SmallGroup(456,2);
// by ID
G=gap.SmallGroup(456,2);
# by ID
G:=PCGroup([5,-2,-3,-2,-2,-19,30,42,2109]);
// Polycyclic
G:=Group<a,b,c|a^8=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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