direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: D8×C17, D4⋊C34, C8⋊1C34, C136⋊5C2, C34.14D4, C68.17C22, (D4×C17)⋊4C2, C4.1(C2×C34), C2.3(D4×C17), SmallGroup(272,25)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8×C17
G = < a,b,c | a17=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D8×C17
►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)
(1 112 62 29 69 125 43 90)(2 113 63 30 70 126 44 91)(3 114 64 31 71 127 45 92)(4 115 65 32 72 128 46 93)(5 116 66 33 73 129 47 94)(6 117 67 34 74 130 48 95)(7 118 68 18 75 131 49 96)(8 119 52 19 76 132 50 97)(9 103 53 20 77 133 51 98)(10 104 54 21 78 134 35 99)(11 105 55 22 79 135 36 100)(12 106 56 23 80 136 37 101)(13 107 57 24 81 120 38 102)(14 108 58 25 82 121 39 86)(15 109 59 26 83 122 40 87)(16 110 60 27 84 123 41 88)(17 111 61 28 85 124 42 89)
(1 90)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 97)(9 98)(10 99)(11 100)(12 101)(13 102)(14 86)(15 87)(16 88)(17 89)(18 75)(19 76)(20 77)(21 78)(22 79)(23 80)(24 81)(25 82)(26 83)(27 84)(28 85)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 104)(36 105)(37 106)(38 107)(39 108)(40 109)(41 110)(42 111)(43 112)(44 113)(45 114)(46 115)(47 116)(48 117)(49 118)(50 119)(51 103)(52 132)(53 133)(54 134)(55 135)(56 136)(57 120)(58 121)(59 122)(60 123)(61 124)(62 125)(63 126)(64 127)(65 128)(66 129)(67 130)(68 131)
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), (1,112,62,29,69,125,43,90)(2,113,63,30,70,126,44,91)(3,114,64,31,71,127,45,92)(4,115,65,32,72,128,46,93)(5,116,66,33,73,129,47,94)(6,117,67,34,74,130,48,95)(7,118,68,18,75,131,49,96)(8,119,52,19,76,132,50,97)(9,103,53,20,77,133,51,98)(10,104,54,21,78,134,35,99)(11,105,55,22,79,135,36,100)(12,106,56,23,80,136,37,101)(13,107,57,24,81,120,38,102)(14,108,58,25,82,121,39,86)(15,109,59,26,83,122,40,87)(16,110,60,27,84,123,41,88)(17,111,61,28,85,124,42,89), (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,100)(12,101)(13,102)(14,86)(15,87)(16,88)(17,89)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,81)(25,82)(26,83)(27,84)(28,85)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,110)(42,111)(43,112)(44,113)(45,114)(46,115)(47,116)(48,117)(49,118)(50,119)(51,103)(52,132)(53,133)(54,134)(55,135)(56,136)(57,120)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131)>;
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), (1,112,62,29,69,125,43,90)(2,113,63,30,70,126,44,91)(3,114,64,31,71,127,45,92)(4,115,65,32,72,128,46,93)(5,116,66,33,73,129,47,94)(6,117,67,34,74,130,48,95)(7,118,68,18,75,131,49,96)(8,119,52,19,76,132,50,97)(9,103,53,20,77,133,51,98)(10,104,54,21,78,134,35,99)(11,105,55,22,79,135,36,100)(12,106,56,23,80,136,37,101)(13,107,57,24,81,120,38,102)(14,108,58,25,82,121,39,86)(15,109,59,26,83,122,40,87)(16,110,60,27,84,123,41,88)(17,111,61,28,85,124,42,89), (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,100)(12,101)(13,102)(14,86)(15,87)(16,88)(17,89)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,81)(25,82)(26,83)(27,84)(28,85)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,110)(42,111)(43,112)(44,113)(45,114)(46,115)(47,116)(48,117)(49,118)(50,119)(51,103)(52,132)(53,133)(54,134)(55,135)(56,136)(57,120)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131) );
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)], [(1,112,62,29,69,125,43,90),(2,113,63,30,70,126,44,91),(3,114,64,31,71,127,45,92),(4,115,65,32,72,128,46,93),(5,116,66,33,73,129,47,94),(6,117,67,34,74,130,48,95),(7,118,68,18,75,131,49,96),(8,119,52,19,76,132,50,97),(9,103,53,20,77,133,51,98),(10,104,54,21,78,134,35,99),(11,105,55,22,79,135,36,100),(12,106,56,23,80,136,37,101),(13,107,57,24,81,120,38,102),(14,108,58,25,82,121,39,86),(15,109,59,26,83,122,40,87),(16,110,60,27,84,123,41,88),(17,111,61,28,85,124,42,89)], [(1,90),(2,91),(3,92),(4,93),(5,94),(6,95),(7,96),(8,97),(9,98),(10,99),(11,100),(12,101),(13,102),(14,86),(15,87),(16,88),(17,89),(18,75),(19,76),(20,77),(21,78),(22,79),(23,80),(24,81),(25,82),(26,83),(27,84),(28,85),(29,69),(30,70),(31,71),(32,72),(33,73),(34,74),(35,104),(36,105),(37,106),(38,107),(39,108),(40,109),(41,110),(42,111),(43,112),(44,113),(45,114),(46,115),(47,116),(48,117),(49,118),(50,119),(51,103),(52,132),(53,133),(54,134),(55,135),(56,136),(57,120),(58,121),(59,122),(60,123),(61,124),(62,125),(63,126),(64,127),(65,128),(66,129),(67,130),(68,131)]])
119 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 8A | 8B | 17A | ··· | 17P | 34A | ··· | 34P | 34Q | ··· | 34AV | 68A | ··· | 68P | 136A | ··· | 136AF |
| order | 1 | 2 | 2 | 2 | 4 | 8 | 8 | 17 | ··· | 17 | 34 | ··· | 34 | 34 | ··· | 34 | 68 | ··· | 68 | 136 | ··· | 136 |
| size | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 |
119 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C17 | C34 | C34 | D4 | D8 | D4×C17 | D8×C17 |
| kernel | D8×C17 | C136 | D4×C17 | D8 | C8 | D4 | C34 | C17 | C2 | C1 |
| # reps | 1 | 1 | 2 | 16 | 16 | 32 | 1 | 2 | 16 | 32 |
Matrix representation of D8×C17 ►in GL2(𝔽137) generated by
| 72 | 0 |
| 0 | 72 |
| 0 | 53 |
| 31 | 106 |
| 106 | 84 |
| 106 | 31 |
G:=sub<GL(2,GF(137))| [72,0,0,72],[0,31,53,106],[106,106,84,31] >;
D8×C17 in GAP, Magma, Sage, TeX
D_8\times C_{17} % in TeX
G:=Group("D8xC17"); // GroupNames label
G:=SmallGroup(272,25);
// by ID
G=gap.SmallGroup(272,25);
# by ID
G:=PCGroup([5,-2,-2,-17,-2,-2,701,4083,2048,58]);
// Polycyclic
G:=Group<a,b,c|a^17=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export