metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D17⋊2C8, C68.3C4, D34.3C4, Dic17.4C22, C17⋊2(C2×C8), C17⋊2C8⋊3C2, C4.3(C17⋊C4), C34.1(C2×C4), (C4×D17).5C2, C2.1(C2×C17⋊C4), SmallGroup(272,29)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — D17⋊2C8 |
Generators and relations for D17⋊2C8
G = < a,b,c | a17=b2=c8=1, bab=a-1, cac-1=a4, cbc-1=a3b >
Smallest permutation representation of D17⋊2C8
►On 136 points
Generators in S136
(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)
(1 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(18 23)(19 22)(20 21)(24 34)(25 33)(26 32)(27 31)(28 30)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(41 45)(42 44)(52 68)(53 67)(54 66)(55 65)(56 64)(57 63)(58 62)(59 61)(69 74)(70 73)(71 72)(75 85)(76 84)(77 83)(78 82)(79 81)(86 93)(87 92)(88 91)(89 90)(94 102)(95 101)(96 100)(97 99)(103 115)(104 114)(105 113)(106 112)(107 111)(108 110)(116 119)(117 118)(120 124)(121 123)(125 136)(126 135)(127 134)(128 133)(129 132)(130 131)
(1 131 52 90 21 118 35 72)(2 127 68 94 22 114 51 76)(3 123 67 98 23 110 50 80)(4 136 66 102 24 106 49 84)(5 132 65 89 25 119 48 71)(6 128 64 93 26 115 47 75)(7 124 63 97 27 111 46 79)(8 120 62 101 28 107 45 83)(9 133 61 88 29 103 44 70)(10 129 60 92 30 116 43 74)(11 125 59 96 31 112 42 78)(12 121 58 100 32 108 41 82)(13 134 57 87 33 104 40 69)(14 130 56 91 34 117 39 73)(15 126 55 95 18 113 38 77)(16 122 54 99 19 109 37 81)(17 135 53 86 20 105 36 85)
G:=sub<Sym(136)| (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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,23)(19,22)(20,21)(24,34)(25,33)(26,32)(27,31)(28,30)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(69,74)(70,73)(71,72)(75,85)(76,84)(77,83)(78,82)(79,81)(86,93)(87,92)(88,91)(89,90)(94,102)(95,101)(96,100)(97,99)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)(116,119)(117,118)(120,124)(121,123)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131), (1,131,52,90,21,118,35,72)(2,127,68,94,22,114,51,76)(3,123,67,98,23,110,50,80)(4,136,66,102,24,106,49,84)(5,132,65,89,25,119,48,71)(6,128,64,93,26,115,47,75)(7,124,63,97,27,111,46,79)(8,120,62,101,28,107,45,83)(9,133,61,88,29,103,44,70)(10,129,60,92,30,116,43,74)(11,125,59,96,31,112,42,78)(12,121,58,100,32,108,41,82)(13,134,57,87,33,104,40,69)(14,130,56,91,34,117,39,73)(15,126,55,95,18,113,38,77)(16,122,54,99,19,109,37,81)(17,135,53,86,20,105,36,85)>;
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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,23)(19,22)(20,21)(24,34)(25,33)(26,32)(27,31)(28,30)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(69,74)(70,73)(71,72)(75,85)(76,84)(77,83)(78,82)(79,81)(86,93)(87,92)(88,91)(89,90)(94,102)(95,101)(96,100)(97,99)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)(116,119)(117,118)(120,124)(121,123)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131), (1,131,52,90,21,118,35,72)(2,127,68,94,22,114,51,76)(3,123,67,98,23,110,50,80)(4,136,66,102,24,106,49,84)(5,132,65,89,25,119,48,71)(6,128,64,93,26,115,47,75)(7,124,63,97,27,111,46,79)(8,120,62,101,28,107,45,83)(9,133,61,88,29,103,44,70)(10,129,60,92,30,116,43,74)(11,125,59,96,31,112,42,78)(12,121,58,100,32,108,41,82)(13,134,57,87,33,104,40,69)(14,130,56,91,34,117,39,73)(15,126,55,95,18,113,38,77)(16,122,54,99,19,109,37,81)(17,135,53,86,20,105,36,85) );
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)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(18,23),(19,22),(20,21),(24,34),(25,33),(26,32),(27,31),(28,30),(35,51),(36,50),(37,49),(38,48),(39,47),(40,46),(41,45),(42,44),(52,68),(53,67),(54,66),(55,65),(56,64),(57,63),(58,62),(59,61),(69,74),(70,73),(71,72),(75,85),(76,84),(77,83),(78,82),(79,81),(86,93),(87,92),(88,91),(89,90),(94,102),(95,101),(96,100),(97,99),(103,115),(104,114),(105,113),(106,112),(107,111),(108,110),(116,119),(117,118),(120,124),(121,123),(125,136),(126,135),(127,134),(128,133),(129,132),(130,131)], [(1,131,52,90,21,118,35,72),(2,127,68,94,22,114,51,76),(3,123,67,98,23,110,50,80),(4,136,66,102,24,106,49,84),(5,132,65,89,25,119,48,71),(6,128,64,93,26,115,47,75),(7,124,63,97,27,111,46,79),(8,120,62,101,28,107,45,83),(9,133,61,88,29,103,44,70),(10,129,60,92,30,116,43,74),(11,125,59,96,31,112,42,78),(12,121,58,100,32,108,41,82),(13,134,57,87,33,104,40,69),(14,130,56,91,34,117,39,73),(15,126,55,95,18,113,38,77),(16,122,54,99,19,109,37,81),(17,135,53,86,20,105,36,85)])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | ··· | 8H | 17A | 17B | 17C | 17D | 34A | 34B | 34C | 34D | 68A | ··· | 68H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 17 | 17 | 17 | 17 | 34 | 34 | 34 | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 17 | 17 | 1 | 1 | 17 | 17 | 17 | ··· | 17 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 |
| type | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C17⋊C4 | C2×C17⋊C4 | D17⋊2C8 |
| kernel | D17⋊2C8 | C17⋊2C8 | C4×D17 | C68 | D34 | D17 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 4 | 4 | 8 |
Matrix representation of D17⋊2C8 ►in GL4(𝔽137) generated by
| 136 | 1 | 0 | 0 |
| 136 | 0 | 1 | 0 |
| 136 | 0 | 0 | 1 |
| 85 | 104 | 33 | 51 |
| 136 | 0 | 0 | 0 |
| 32 | 104 | 86 | 1 |
| 5 | 47 | 3 | 51 |
| 110 | 76 | 114 | 31 |
| 10 | 0 | 127 | 35 |
| 84 | 98 | 14 | 79 |
| 106 | 76 | 131 | 79 |
| 63 | 84 | 12 | 35 |
G:=sub<GL(4,GF(137))| [136,136,136,85,1,0,0,104,0,1,0,33,0,0,1,51],[136,32,5,110,0,104,47,76,0,86,3,114,0,1,51,31],[10,84,106,63,0,98,76,84,127,14,131,12,35,79,79,35] >;
D17⋊2C8 in GAP, Magma, Sage, TeX
D_{17}\rtimes_2C_8 % in TeX
G:=Group("D17:2C8"); // GroupNames label
G:=SmallGroup(272,29);
// by ID
G=gap.SmallGroup(272,29);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,20,46,42,5204,1614]);
// Polycyclic
G:=Group<a,b,c|a^17=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^4,c*b*c^-1=a^3*b>;
// generators/relations