metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊D17, C34.9D4, C4.3D34, C17⋊3SD16, D68.2C2, C68.3C22, C17⋊3C8⋊3C2, (Q8×C17)⋊1C2, C2.6(C17⋊D4), SmallGroup(272,17)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊D17
G = < a,b,c,d | a4=c17=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=c-1 >
Smallest permutation representation of Q8⋊D17
►On 136 points
Generators in S136
(1 55 22 50)(2 56 23 51)(3 57 24 35)(4 58 25 36)(5 59 26 37)(6 60 27 38)(7 61 28 39)(8 62 29 40)(9 63 30 41)(10 64 31 42)(11 65 32 43)(12 66 33 44)(13 67 34 45)(14 68 18 46)(15 52 19 47)(16 53 20 48)(17 54 21 49)(69 117 101 129)(70 118 102 130)(71 119 86 131)(72 103 87 132)(73 104 88 133)(74 105 89 134)(75 106 90 135)(76 107 91 136)(77 108 92 120)(78 109 93 121)(79 110 94 122)(80 111 95 123)(81 112 96 124)(82 113 97 125)(83 114 98 126)(84 115 99 127)(85 116 100 128)
(1 92 22 77)(2 93 23 78)(3 94 24 79)(4 95 25 80)(5 96 26 81)(6 97 27 82)(7 98 28 83)(8 99 29 84)(9 100 30 85)(10 101 31 69)(11 102 32 70)(12 86 33 71)(13 87 34 72)(14 88 18 73)(15 89 19 74)(16 90 20 75)(17 91 21 76)(35 122 57 110)(36 123 58 111)(37 124 59 112)(38 125 60 113)(39 126 61 114)(40 127 62 115)(41 128 63 116)(42 129 64 117)(43 130 65 118)(44 131 66 119)(45 132 67 103)(46 133 68 104)(47 134 52 105)(48 135 53 106)(49 136 54 107)(50 120 55 108)(51 121 56 109)
(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 25)(19 24)(20 23)(21 22)(26 34)(27 33)(28 32)(29 31)(35 52)(36 68)(37 67)(38 66)(39 65)(40 64)(41 63)(42 62)(43 61)(44 60)(45 59)(46 58)(47 57)(48 56)(49 55)(50 54)(51 53)(69 115)(70 114)(71 113)(72 112)(73 111)(74 110)(75 109)(76 108)(77 107)(78 106)(79 105)(80 104)(81 103)(82 119)(83 118)(84 117)(85 116)(86 125)(87 124)(88 123)(89 122)(90 121)(91 120)(92 136)(93 135)(94 134)(95 133)(96 132)(97 131)(98 130)(99 129)(100 128)(101 127)(102 126)
G:=sub<Sym(136)| (1,55,22,50)(2,56,23,51)(3,57,24,35)(4,58,25,36)(5,59,26,37)(6,60,27,38)(7,61,28,39)(8,62,29,40)(9,63,30,41)(10,64,31,42)(11,65,32,43)(12,66,33,44)(13,67,34,45)(14,68,18,46)(15,52,19,47)(16,53,20,48)(17,54,21,49)(69,117,101,129)(70,118,102,130)(71,119,86,131)(72,103,87,132)(73,104,88,133)(74,105,89,134)(75,106,90,135)(76,107,91,136)(77,108,92,120)(78,109,93,121)(79,110,94,122)(80,111,95,123)(81,112,96,124)(82,113,97,125)(83,114,98,126)(84,115,99,127)(85,116,100,128), (1,92,22,77)(2,93,23,78)(3,94,24,79)(4,95,25,80)(5,96,26,81)(6,97,27,82)(7,98,28,83)(8,99,29,84)(9,100,30,85)(10,101,31,69)(11,102,32,70)(12,86,33,71)(13,87,34,72)(14,88,18,73)(15,89,19,74)(16,90,20,75)(17,91,21,76)(35,122,57,110)(36,123,58,111)(37,124,59,112)(38,125,60,113)(39,126,61,114)(40,127,62,115)(41,128,63,116)(42,129,64,117)(43,130,65,118)(44,131,66,119)(45,132,67,103)(46,133,68,104)(47,134,52,105)(48,135,53,106)(49,136,54,107)(50,120,55,108)(51,121,56,109), (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)(18,25)(19,24)(20,23)(21,22)(26,34)(27,33)(28,32)(29,31)(35,52)(36,68)(37,67)(38,66)(39,65)(40,64)(41,63)(42,62)(43,61)(44,60)(45,59)(46,58)(47,57)(48,56)(49,55)(50,54)(51,53)(69,115)(70,114)(71,113)(72,112)(73,111)(74,110)(75,109)(76,108)(77,107)(78,106)(79,105)(80,104)(81,103)(82,119)(83,118)(84,117)(85,116)(86,125)(87,124)(88,123)(89,122)(90,121)(91,120)(92,136)(93,135)(94,134)(95,133)(96,132)(97,131)(98,130)(99,129)(100,128)(101,127)(102,126)>;
G:=Group( (1,55,22,50)(2,56,23,51)(3,57,24,35)(4,58,25,36)(5,59,26,37)(6,60,27,38)(7,61,28,39)(8,62,29,40)(9,63,30,41)(10,64,31,42)(11,65,32,43)(12,66,33,44)(13,67,34,45)(14,68,18,46)(15,52,19,47)(16,53,20,48)(17,54,21,49)(69,117,101,129)(70,118,102,130)(71,119,86,131)(72,103,87,132)(73,104,88,133)(74,105,89,134)(75,106,90,135)(76,107,91,136)(77,108,92,120)(78,109,93,121)(79,110,94,122)(80,111,95,123)(81,112,96,124)(82,113,97,125)(83,114,98,126)(84,115,99,127)(85,116,100,128), (1,92,22,77)(2,93,23,78)(3,94,24,79)(4,95,25,80)(5,96,26,81)(6,97,27,82)(7,98,28,83)(8,99,29,84)(9,100,30,85)(10,101,31,69)(11,102,32,70)(12,86,33,71)(13,87,34,72)(14,88,18,73)(15,89,19,74)(16,90,20,75)(17,91,21,76)(35,122,57,110)(36,123,58,111)(37,124,59,112)(38,125,60,113)(39,126,61,114)(40,127,62,115)(41,128,63,116)(42,129,64,117)(43,130,65,118)(44,131,66,119)(45,132,67,103)(46,133,68,104)(47,134,52,105)(48,135,53,106)(49,136,54,107)(50,120,55,108)(51,121,56,109), (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)(18,25)(19,24)(20,23)(21,22)(26,34)(27,33)(28,32)(29,31)(35,52)(36,68)(37,67)(38,66)(39,65)(40,64)(41,63)(42,62)(43,61)(44,60)(45,59)(46,58)(47,57)(48,56)(49,55)(50,54)(51,53)(69,115)(70,114)(71,113)(72,112)(73,111)(74,110)(75,109)(76,108)(77,107)(78,106)(79,105)(80,104)(81,103)(82,119)(83,118)(84,117)(85,116)(86,125)(87,124)(88,123)(89,122)(90,121)(91,120)(92,136)(93,135)(94,134)(95,133)(96,132)(97,131)(98,130)(99,129)(100,128)(101,127)(102,126) );
G=PermutationGroup([[(1,55,22,50),(2,56,23,51),(3,57,24,35),(4,58,25,36),(5,59,26,37),(6,60,27,38),(7,61,28,39),(8,62,29,40),(9,63,30,41),(10,64,31,42),(11,65,32,43),(12,66,33,44),(13,67,34,45),(14,68,18,46),(15,52,19,47),(16,53,20,48),(17,54,21,49),(69,117,101,129),(70,118,102,130),(71,119,86,131),(72,103,87,132),(73,104,88,133),(74,105,89,134),(75,106,90,135),(76,107,91,136),(77,108,92,120),(78,109,93,121),(79,110,94,122),(80,111,95,123),(81,112,96,124),(82,113,97,125),(83,114,98,126),(84,115,99,127),(85,116,100,128)], [(1,92,22,77),(2,93,23,78),(3,94,24,79),(4,95,25,80),(5,96,26,81),(6,97,27,82),(7,98,28,83),(8,99,29,84),(9,100,30,85),(10,101,31,69),(11,102,32,70),(12,86,33,71),(13,87,34,72),(14,88,18,73),(15,89,19,74),(16,90,20,75),(17,91,21,76),(35,122,57,110),(36,123,58,111),(37,124,59,112),(38,125,60,113),(39,126,61,114),(40,127,62,115),(41,128,63,116),(42,129,64,117),(43,130,65,118),(44,131,66,119),(45,132,67,103),(46,133,68,104),(47,134,52,105),(48,135,53,106),(49,136,54,107),(50,120,55,108),(51,121,56,109)], [(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),(18,25),(19,24),(20,23),(21,22),(26,34),(27,33),(28,32),(29,31),(35,52),(36,68),(37,67),(38,66),(39,65),(40,64),(41,63),(42,62),(43,61),(44,60),(45,59),(46,58),(47,57),(48,56),(49,55),(50,54),(51,53),(69,115),(70,114),(71,113),(72,112),(73,111),(74,110),(75,109),(76,108),(77,107),(78,106),(79,105),(80,104),(81,103),(82,119),(83,118),(84,117),(85,116),(86,125),(87,124),(88,123),(89,122),(90,121),(91,120),(92,136),(93,135),(94,134),(95,133),(96,132),(97,131),(98,130),(99,129),(100,128),(101,127),(102,126)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 17A | ··· | 17H | 34A | ··· | 34H | 68A | ··· | 68X |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 68 | 2 | 4 | 34 | 34 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | D4 | SD16 | D17 | D34 | C17⋊D4 | Q8⋊D17 |
| kernel | Q8⋊D17 | C17⋊3C8 | D68 | Q8×C17 | C34 | C17 | Q8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 8 | 8 | 16 | 8 |
Matrix representation of Q8⋊D17 ►in GL4(𝔽137) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 28 |
| 0 | 0 | 88 | 136 |
| 136 | 0 | 0 | 0 |
| 0 | 136 | 0 | 0 |
| 0 | 0 | 51 | 29 |
| 0 | 0 | 52 | 86 |
| 0 | 1 | 0 | 0 |
| 136 | 34 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 88 | 136 |
G:=sub<GL(4,GF(137))| [1,0,0,0,0,1,0,0,0,0,1,88,0,0,28,136],[136,0,0,0,0,136,0,0,0,0,51,52,0,0,29,86],[0,136,0,0,1,34,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,88,0,0,0,136] >;
Q8⋊D17 in GAP, Magma, Sage, TeX
Q_8\rtimes D_{17} % in TeX
G:=Group("Q8:D17"); // GroupNames label
G:=SmallGroup(272,17);
// by ID
G=gap.SmallGroup(272,17);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,61,46,182,97,42,6404]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^17=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations
Export