direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: SD16×C17, Q8⋊C34, C8⋊2C34, D4.C34, C136⋊6C2, C34.15D4, C68.18C22, C4.2(C2×C34), (Q8×C17)⋊4C2, C2.4(D4×C17), (D4×C17).2C2, SmallGroup(272,26)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16×C17
G = < a,b,c | a17=b8=c2=1, ab=ba, ac=ca, cbc=b3 >
Smallest permutation representation of SD16×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 76 61 117 32 136 49 102)(2 77 62 118 33 120 50 86)(3 78 63 119 34 121 51 87)(4 79 64 103 18 122 35 88)(5 80 65 104 19 123 36 89)(6 81 66 105 20 124 37 90)(7 82 67 106 21 125 38 91)(8 83 68 107 22 126 39 92)(9 84 52 108 23 127 40 93)(10 85 53 109 24 128 41 94)(11 69 54 110 25 129 42 95)(12 70 55 111 26 130 43 96)(13 71 56 112 27 131 44 97)(14 72 57 113 28 132 45 98)(15 73 58 114 29 133 46 99)(16 74 59 115 30 134 47 100)(17 75 60 116 31 135 48 101)
(35 64)(36 65)(37 66)(38 67)(39 68)(40 52)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)(49 61)(50 62)(51 63)(69 110)(70 111)(71 112)(72 113)(73 114)(74 115)(75 116)(76 117)(77 118)(78 119)(79 103)(80 104)(81 105)(82 106)(83 107)(84 108)(85 109)(86 120)(87 121)(88 122)(89 123)(90 124)(91 125)(92 126)(93 127)(94 128)(95 129)(96 130)(97 131)(98 132)(99 133)(100 134)(101 135)(102 136)
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), (1,76,61,117,32,136,49,102)(2,77,62,118,33,120,50,86)(3,78,63,119,34,121,51,87)(4,79,64,103,18,122,35,88)(5,80,65,104,19,123,36,89)(6,81,66,105,20,124,37,90)(7,82,67,106,21,125,38,91)(8,83,68,107,22,126,39,92)(9,84,52,108,23,127,40,93)(10,85,53,109,24,128,41,94)(11,69,54,110,25,129,42,95)(12,70,55,111,26,130,43,96)(13,71,56,112,27,131,44,97)(14,72,57,113,28,132,45,98)(15,73,58,114,29,133,46,99)(16,74,59,115,30,134,47,100)(17,75,60,116,31,135,48,101), (35,64)(36,65)(37,66)(38,67)(39,68)(40,52)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60)(49,61)(50,62)(51,63)(69,110)(70,111)(71,112)(72,113)(73,114)(74,115)(75,116)(76,117)(77,118)(78,119)(79,103)(80,104)(81,105)(82,106)(83,107)(84,108)(85,109)(86,120)(87,121)(88,122)(89,123)(90,124)(91,125)(92,126)(93,127)(94,128)(95,129)(96,130)(97,131)(98,132)(99,133)(100,134)(101,135)(102,136)>;
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), (1,76,61,117,32,136,49,102)(2,77,62,118,33,120,50,86)(3,78,63,119,34,121,51,87)(4,79,64,103,18,122,35,88)(5,80,65,104,19,123,36,89)(6,81,66,105,20,124,37,90)(7,82,67,106,21,125,38,91)(8,83,68,107,22,126,39,92)(9,84,52,108,23,127,40,93)(10,85,53,109,24,128,41,94)(11,69,54,110,25,129,42,95)(12,70,55,111,26,130,43,96)(13,71,56,112,27,131,44,97)(14,72,57,113,28,132,45,98)(15,73,58,114,29,133,46,99)(16,74,59,115,30,134,47,100)(17,75,60,116,31,135,48,101), (35,64)(36,65)(37,66)(38,67)(39,68)(40,52)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60)(49,61)(50,62)(51,63)(69,110)(70,111)(71,112)(72,113)(73,114)(74,115)(75,116)(76,117)(77,118)(78,119)(79,103)(80,104)(81,105)(82,106)(83,107)(84,108)(85,109)(86,120)(87,121)(88,122)(89,123)(90,124)(91,125)(92,126)(93,127)(94,128)(95,129)(96,130)(97,131)(98,132)(99,133)(100,134)(101,135)(102,136) );
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)], [(1,76,61,117,32,136,49,102),(2,77,62,118,33,120,50,86),(3,78,63,119,34,121,51,87),(4,79,64,103,18,122,35,88),(5,80,65,104,19,123,36,89),(6,81,66,105,20,124,37,90),(7,82,67,106,21,125,38,91),(8,83,68,107,22,126,39,92),(9,84,52,108,23,127,40,93),(10,85,53,109,24,128,41,94),(11,69,54,110,25,129,42,95),(12,70,55,111,26,130,43,96),(13,71,56,112,27,131,44,97),(14,72,57,113,28,132,45,98),(15,73,58,114,29,133,46,99),(16,74,59,115,30,134,47,100),(17,75,60,116,31,135,48,101)], [(35,64),(36,65),(37,66),(38,67),(39,68),(40,52),(41,53),(42,54),(43,55),(44,56),(45,57),(46,58),(47,59),(48,60),(49,61),(50,62),(51,63),(69,110),(70,111),(71,112),(72,113),(73,114),(74,115),(75,116),(76,117),(77,118),(78,119),(79,103),(80,104),(81,105),(82,106),(83,107),(84,108),(85,109),(86,120),(87,121),(88,122),(89,123),(90,124),(91,125),(92,126),(93,127),(94,128),(95,129),(96,130),(97,131),(98,132),(99,133),(100,134),(101,135),(102,136)]])
119 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 17A | ··· | 17P | 34A | ··· | 34P | 34Q | ··· | 34AF | 68A | ··· | 68P | 68Q | ··· | 68AF | 136A | ··· | 136AF |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 17 | ··· | 17 | 34 | ··· | 34 | 34 | ··· | 34 | 68 | ··· | 68 | 68 | ··· | 68 | 136 | ··· | 136 |
| size | 1 | 1 | 4 | 2 | 4 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
119 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C17 | C34 | C34 | C34 | D4 | SD16 | D4×C17 | SD16×C17 |
| kernel | SD16×C17 | C136 | D4×C17 | Q8×C17 | SD16 | C8 | D4 | Q8 | C34 | C17 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 16 | 16 | 16 | 16 | 1 | 2 | 16 | 32 |
Matrix representation of SD16×C17 ►in GL2(𝔽137) generated by
| 38 | 0 |
| 0 | 38 |
| 43 | 94 |
| 43 | 43 |
| 1 | 0 |
| 0 | 136 |
G:=sub<GL(2,GF(137))| [38,0,0,38],[43,43,94,43],[1,0,0,136] >;
SD16×C17 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\times C_{17} % in TeX
G:=Group("SD16xC17"); // GroupNames label
G:=SmallGroup(272,26);
// by ID
G=gap.SmallGroup(272,26);
# by ID
G:=PCGroup([5,-2,-2,-17,-2,-2,680,701,4083,2048,58]);
// 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^3>;
// generators/relations
Export