direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8×D17, C4.6D34, Dic34⋊4C2, C68.6C22, C34.7C23, D34.9C22, Dic17.3C22, C17⋊2(C2×Q8), (Q8×C17)⋊2C2, (C4×D17).1C2, C2.8(C22×D17), SmallGroup(272,42)
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=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of Q8×D17
►On 136 points
Generators in S136
(1 53 27 43)(2 54 28 44)(3 55 29 45)(4 56 30 46)(5 57 31 47)(6 58 32 48)(7 59 33 49)(8 60 34 50)(9 61 18 51)(10 62 19 35)(11 63 20 36)(12 64 21 37)(13 65 22 38)(14 66 23 39)(15 67 24 40)(16 68 25 41)(17 52 26 42)(69 118 96 132)(70 119 97 133)(71 103 98 134)(72 104 99 135)(73 105 100 136)(74 106 101 120)(75 107 102 121)(76 108 86 122)(77 109 87 123)(78 110 88 124)(79 111 89 125)(80 112 90 126)(81 113 91 127)(82 114 92 128)(83 115 93 129)(84 116 94 130)(85 117 95 131)
(1 100 27 73)(2 101 28 74)(3 102 29 75)(4 86 30 76)(5 87 31 77)(6 88 32 78)(7 89 33 79)(8 90 34 80)(9 91 18 81)(10 92 19 82)(11 93 20 83)(12 94 21 84)(13 95 22 85)(14 96 23 69)(15 97 24 70)(16 98 25 71)(17 99 26 72)(35 128 62 114)(36 129 63 115)(37 130 64 116)(38 131 65 117)(39 132 66 118)(40 133 67 119)(41 134 68 103)(42 135 52 104)(43 136 53 105)(44 120 54 106)(45 121 55 107)(46 122 56 108)(47 123 57 109)(48 124 58 110)(49 125 59 111)(50 126 60 112)(51 127 61 113)
(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 26)(2 25)(3 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 18)(10 34)(11 33)(12 32)(13 31)(14 30)(15 29)(16 28)(17 27)(35 60)(36 59)(37 58)(38 57)(39 56)(40 55)(41 54)(42 53)(43 52)(44 68)(45 67)(46 66)(47 65)(48 64)(49 63)(50 62)(51 61)(69 86)(70 102)(71 101)(72 100)(73 99)(74 98)(75 97)(76 96)(77 95)(78 94)(79 93)(80 92)(81 91)(82 90)(83 89)(84 88)(85 87)(103 120)(104 136)(105 135)(106 134)(107 133)(108 132)(109 131)(110 130)(111 129)(112 128)(113 127)(114 126)(115 125)(116 124)(117 123)(118 122)(119 121)
G:=sub<Sym(136)| (1,53,27,43)(2,54,28,44)(3,55,29,45)(4,56,30,46)(5,57,31,47)(6,58,32,48)(7,59,33,49)(8,60,34,50)(9,61,18,51)(10,62,19,35)(11,63,20,36)(12,64,21,37)(13,65,22,38)(14,66,23,39)(15,67,24,40)(16,68,25,41)(17,52,26,42)(69,118,96,132)(70,119,97,133)(71,103,98,134)(72,104,99,135)(73,105,100,136)(74,106,101,120)(75,107,102,121)(76,108,86,122)(77,109,87,123)(78,110,88,124)(79,111,89,125)(80,112,90,126)(81,113,91,127)(82,114,92,128)(83,115,93,129)(84,116,94,130)(85,117,95,131), (1,100,27,73)(2,101,28,74)(3,102,29,75)(4,86,30,76)(5,87,31,77)(6,88,32,78)(7,89,33,79)(8,90,34,80)(9,91,18,81)(10,92,19,82)(11,93,20,83)(12,94,21,84)(13,95,22,85)(14,96,23,69)(15,97,24,70)(16,98,25,71)(17,99,26,72)(35,128,62,114)(36,129,63,115)(37,130,64,116)(38,131,65,117)(39,132,66,118)(40,133,67,119)(41,134,68,103)(42,135,52,104)(43,136,53,105)(44,120,54,106)(45,121,55,107)(46,122,56,108)(47,123,57,109)(48,124,58,110)(49,125,59,111)(50,126,60,112)(51,127,61,113), (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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,27)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,68)(45,67)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(69,86)(70,102)(71,101)(72,100)(73,99)(74,98)(75,97)(76,96)(77,95)(78,94)(79,93)(80,92)(81,91)(82,90)(83,89)(84,88)(85,87)(103,120)(104,136)(105,135)(106,134)(107,133)(108,132)(109,131)(110,130)(111,129)(112,128)(113,127)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121)>;
G:=Group( (1,53,27,43)(2,54,28,44)(3,55,29,45)(4,56,30,46)(5,57,31,47)(6,58,32,48)(7,59,33,49)(8,60,34,50)(9,61,18,51)(10,62,19,35)(11,63,20,36)(12,64,21,37)(13,65,22,38)(14,66,23,39)(15,67,24,40)(16,68,25,41)(17,52,26,42)(69,118,96,132)(70,119,97,133)(71,103,98,134)(72,104,99,135)(73,105,100,136)(74,106,101,120)(75,107,102,121)(76,108,86,122)(77,109,87,123)(78,110,88,124)(79,111,89,125)(80,112,90,126)(81,113,91,127)(82,114,92,128)(83,115,93,129)(84,116,94,130)(85,117,95,131), (1,100,27,73)(2,101,28,74)(3,102,29,75)(4,86,30,76)(5,87,31,77)(6,88,32,78)(7,89,33,79)(8,90,34,80)(9,91,18,81)(10,92,19,82)(11,93,20,83)(12,94,21,84)(13,95,22,85)(14,96,23,69)(15,97,24,70)(16,98,25,71)(17,99,26,72)(35,128,62,114)(36,129,63,115)(37,130,64,116)(38,131,65,117)(39,132,66,118)(40,133,67,119)(41,134,68,103)(42,135,52,104)(43,136,53,105)(44,120,54,106)(45,121,55,107)(46,122,56,108)(47,123,57,109)(48,124,58,110)(49,125,59,111)(50,126,60,112)(51,127,61,113), (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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,27)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,68)(45,67)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(69,86)(70,102)(71,101)(72,100)(73,99)(74,98)(75,97)(76,96)(77,95)(78,94)(79,93)(80,92)(81,91)(82,90)(83,89)(84,88)(85,87)(103,120)(104,136)(105,135)(106,134)(107,133)(108,132)(109,131)(110,130)(111,129)(112,128)(113,127)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121) );
G=PermutationGroup([[(1,53,27,43),(2,54,28,44),(3,55,29,45),(4,56,30,46),(5,57,31,47),(6,58,32,48),(7,59,33,49),(8,60,34,50),(9,61,18,51),(10,62,19,35),(11,63,20,36),(12,64,21,37),(13,65,22,38),(14,66,23,39),(15,67,24,40),(16,68,25,41),(17,52,26,42),(69,118,96,132),(70,119,97,133),(71,103,98,134),(72,104,99,135),(73,105,100,136),(74,106,101,120),(75,107,102,121),(76,108,86,122),(77,109,87,123),(78,110,88,124),(79,111,89,125),(80,112,90,126),(81,113,91,127),(82,114,92,128),(83,115,93,129),(84,116,94,130),(85,117,95,131)], [(1,100,27,73),(2,101,28,74),(3,102,29,75),(4,86,30,76),(5,87,31,77),(6,88,32,78),(7,89,33,79),(8,90,34,80),(9,91,18,81),(10,92,19,82),(11,93,20,83),(12,94,21,84),(13,95,22,85),(14,96,23,69),(15,97,24,70),(16,98,25,71),(17,99,26,72),(35,128,62,114),(36,129,63,115),(37,130,64,116),(38,131,65,117),(39,132,66,118),(40,133,67,119),(41,134,68,103),(42,135,52,104),(43,136,53,105),(44,120,54,106),(45,121,55,107),(46,122,56,108),(47,123,57,109),(48,124,58,110),(49,125,59,111),(50,126,60,112),(51,127,61,113)], [(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)], [(1,26),(2,25),(3,24),(4,23),(5,22),(6,21),(7,20),(8,19),(9,18),(10,34),(11,33),(12,32),(13,31),(14,30),(15,29),(16,28),(17,27),(35,60),(36,59),(37,58),(38,57),(39,56),(40,55),(41,54),(42,53),(43,52),(44,68),(45,67),(46,66),(47,65),(48,64),(49,63),(50,62),(51,61),(69,86),(70,102),(71,101),(72,100),(73,99),(74,98),(75,97),(76,96),(77,95),(78,94),(79,93),(80,92),(81,91),(82,90),(83,89),(84,88),(85,87),(103,120),(104,136),(105,135),(106,134),(107,133),(108,132),(109,131),(110,130),(111,129),(112,128),(113,127),(114,126),(115,125),(116,124),(117,123),(118,122),(119,121)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 17A | ··· | 17H | 34A | ··· | 34H | 68A | ··· | 68X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 17 | 17 | 2 | 2 | 2 | 34 | 34 | 34 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | Q8 | D17 | D34 | Q8×D17 |
| kernel | Q8×D17 | Dic34 | C4×D17 | Q8×C17 | D17 | Q8 | C4 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 8 | 24 | 8 |
Matrix representation of Q8×D17 ►in GL4(𝔽137) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 66 | 44 |
| 0 | 0 | 66 | 71 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 10 | 48 |
| 0 | 0 | 35 | 127 |
| 40 | 1 | 0 | 0 |
| 126 | 34 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 21 | 104 | 0 | 0 |
| 59 | 116 | 0 | 0 |
| 0 | 0 | 136 | 0 |
| 0 | 0 | 0 | 136 |
G:=sub<GL(4,GF(137))| [1,0,0,0,0,1,0,0,0,0,66,66,0,0,44,71],[1,0,0,0,0,1,0,0,0,0,10,35,0,0,48,127],[40,126,0,0,1,34,0,0,0,0,1,0,0,0,0,1],[21,59,0,0,104,116,0,0,0,0,136,0,0,0,0,136] >;
Q8×D17 in GAP, Magma, Sage, TeX
Q_8\times D_{17} % in TeX
G:=Group("Q8xD17"); // GroupNames label
G:=SmallGroup(272,42);
// by ID
G=gap.SmallGroup(272,42);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,46,97,42,6404]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^17=d^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export