metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊D17, C17⋊2D8, D68⋊2C2, C4.1D34, C34.7D4, C68.1C22, C17⋊3C8⋊1C2, (D4×C17)⋊1C2, C2.4(C17⋊D4), SmallGroup(272,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊D17
G = < a,b,c,d | a4=b2=c17=d2=1, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=c-1 >
Smallest permutation representation of D4⋊D17
►On 136 points
Generators in S136
(1 55 32 44)(2 56 33 45)(3 57 34 46)(4 58 18 47)(5 59 19 48)(6 60 20 49)(7 61 21 50)(8 62 22 51)(9 63 23 35)(10 64 24 36)(11 65 25 37)(12 66 26 38)(13 67 27 39)(14 68 28 40)(15 52 29 41)(16 53 30 42)(17 54 31 43)(69 109 94 130)(70 110 95 131)(71 111 96 132)(72 112 97 133)(73 113 98 134)(74 114 99 135)(75 115 100 136)(76 116 101 120)(77 117 102 121)(78 118 86 122)(79 119 87 123)(80 103 88 124)(81 104 89 125)(82 105 90 126)(83 106 91 127)(84 107 92 128)(85 108 93 129)
(1 123)(2 124)(3 125)(4 126)(5 127)(6 128)(7 129)(8 130)(9 131)(10 132)(11 133)(12 134)(13 135)(14 136)(15 120)(16 121)(17 122)(18 105)(19 106)(20 107)(21 108)(22 109)(23 110)(24 111)(25 112)(26 113)(27 114)(28 115)(29 116)(30 117)(31 118)(32 119)(33 103)(34 104)(35 70)(36 71)(37 72)(38 73)(39 74)(40 75)(41 76)(42 77)(43 78)(44 79)(45 80)(46 81)(47 82)(48 83)(49 84)(50 85)(51 69)(52 101)(53 102)(54 86)(55 87)(56 88)(57 89)(58 90)(59 91)(60 92)(61 93)(62 94)(63 95)(64 96)(65 97)(66 98)(67 99)(68 100)
(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 28)(19 27)(20 26)(21 25)(22 24)(29 34)(30 33)(31 32)(35 63)(36 62)(37 61)(38 60)(39 59)(40 58)(41 57)(42 56)(43 55)(44 54)(45 53)(46 52)(47 68)(48 67)(49 66)(50 65)(51 64)(69 111)(70 110)(71 109)(72 108)(73 107)(74 106)(75 105)(76 104)(77 103)(78 119)(79 118)(80 117)(81 116)(82 115)(83 114)(84 113)(85 112)(86 123)(87 122)(88 121)(89 120)(90 136)(91 135)(92 134)(93 133)(94 132)(95 131)(96 130)(97 129)(98 128)(99 127)(100 126)(101 125)(102 124)
G:=sub<Sym(136)| (1,55,32,44)(2,56,33,45)(3,57,34,46)(4,58,18,47)(5,59,19,48)(6,60,20,49)(7,61,21,50)(8,62,22,51)(9,63,23,35)(10,64,24,36)(11,65,25,37)(12,66,26,38)(13,67,27,39)(14,68,28,40)(15,52,29,41)(16,53,30,42)(17,54,31,43)(69,109,94,130)(70,110,95,131)(71,111,96,132)(72,112,97,133)(73,113,98,134)(74,114,99,135)(75,115,100,136)(76,116,101,120)(77,117,102,121)(78,118,86,122)(79,119,87,123)(80,103,88,124)(81,104,89,125)(82,105,90,126)(83,106,91,127)(84,107,92,128)(85,108,93,129), (1,123)(2,124)(3,125)(4,126)(5,127)(6,128)(7,129)(8,130)(9,131)(10,132)(11,133)(12,134)(13,135)(14,136)(15,120)(16,121)(17,122)(18,105)(19,106)(20,107)(21,108)(22,109)(23,110)(24,111)(25,112)(26,113)(27,114)(28,115)(29,116)(30,117)(31,118)(32,119)(33,103)(34,104)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,84)(50,85)(51,69)(52,101)(53,102)(54,86)(55,87)(56,88)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100), (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)(18,28)(19,27)(20,26)(21,25)(22,24)(29,34)(30,33)(31,32)(35,63)(36,62)(37,61)(38,60)(39,59)(40,58)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,68)(48,67)(49,66)(50,65)(51,64)(69,111)(70,110)(71,109)(72,108)(73,107)(74,106)(75,105)(76,104)(77,103)(78,119)(79,118)(80,117)(81,116)(82,115)(83,114)(84,113)(85,112)(86,123)(87,122)(88,121)(89,120)(90,136)(91,135)(92,134)(93,133)(94,132)(95,131)(96,130)(97,129)(98,128)(99,127)(100,126)(101,125)(102,124)>;
G:=Group( (1,55,32,44)(2,56,33,45)(3,57,34,46)(4,58,18,47)(5,59,19,48)(6,60,20,49)(7,61,21,50)(8,62,22,51)(9,63,23,35)(10,64,24,36)(11,65,25,37)(12,66,26,38)(13,67,27,39)(14,68,28,40)(15,52,29,41)(16,53,30,42)(17,54,31,43)(69,109,94,130)(70,110,95,131)(71,111,96,132)(72,112,97,133)(73,113,98,134)(74,114,99,135)(75,115,100,136)(76,116,101,120)(77,117,102,121)(78,118,86,122)(79,119,87,123)(80,103,88,124)(81,104,89,125)(82,105,90,126)(83,106,91,127)(84,107,92,128)(85,108,93,129), (1,123)(2,124)(3,125)(4,126)(5,127)(6,128)(7,129)(8,130)(9,131)(10,132)(11,133)(12,134)(13,135)(14,136)(15,120)(16,121)(17,122)(18,105)(19,106)(20,107)(21,108)(22,109)(23,110)(24,111)(25,112)(26,113)(27,114)(28,115)(29,116)(30,117)(31,118)(32,119)(33,103)(34,104)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,84)(50,85)(51,69)(52,101)(53,102)(54,86)(55,87)(56,88)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100), (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)(18,28)(19,27)(20,26)(21,25)(22,24)(29,34)(30,33)(31,32)(35,63)(36,62)(37,61)(38,60)(39,59)(40,58)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,68)(48,67)(49,66)(50,65)(51,64)(69,111)(70,110)(71,109)(72,108)(73,107)(74,106)(75,105)(76,104)(77,103)(78,119)(79,118)(80,117)(81,116)(82,115)(83,114)(84,113)(85,112)(86,123)(87,122)(88,121)(89,120)(90,136)(91,135)(92,134)(93,133)(94,132)(95,131)(96,130)(97,129)(98,128)(99,127)(100,126)(101,125)(102,124) );
G=PermutationGroup([[(1,55,32,44),(2,56,33,45),(3,57,34,46),(4,58,18,47),(5,59,19,48),(6,60,20,49),(7,61,21,50),(8,62,22,51),(9,63,23,35),(10,64,24,36),(11,65,25,37),(12,66,26,38),(13,67,27,39),(14,68,28,40),(15,52,29,41),(16,53,30,42),(17,54,31,43),(69,109,94,130),(70,110,95,131),(71,111,96,132),(72,112,97,133),(73,113,98,134),(74,114,99,135),(75,115,100,136),(76,116,101,120),(77,117,102,121),(78,118,86,122),(79,119,87,123),(80,103,88,124),(81,104,89,125),(82,105,90,126),(83,106,91,127),(84,107,92,128),(85,108,93,129)], [(1,123),(2,124),(3,125),(4,126),(5,127),(6,128),(7,129),(8,130),(9,131),(10,132),(11,133),(12,134),(13,135),(14,136),(15,120),(16,121),(17,122),(18,105),(19,106),(20,107),(21,108),(22,109),(23,110),(24,111),(25,112),(26,113),(27,114),(28,115),(29,116),(30,117),(31,118),(32,119),(33,103),(34,104),(35,70),(36,71),(37,72),(38,73),(39,74),(40,75),(41,76),(42,77),(43,78),(44,79),(45,80),(46,81),(47,82),(48,83),(49,84),(50,85),(51,69),(52,101),(53,102),(54,86),(55,87),(56,88),(57,89),(58,90),(59,91),(60,92),(61,93),(62,94),(63,95),(64,96),(65,97),(66,98),(67,99),(68,100)], [(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),(18,28),(19,27),(20,26),(21,25),(22,24),(29,34),(30,33),(31,32),(35,63),(36,62),(37,61),(38,60),(39,59),(40,58),(41,57),(42,56),(43,55),(44,54),(45,53),(46,52),(47,68),(48,67),(49,66),(50,65),(51,64),(69,111),(70,110),(71,109),(72,108),(73,107),(74,106),(75,105),(76,104),(77,103),(78,119),(79,118),(80,117),(81,116),(82,115),(83,114),(84,113),(85,112),(86,123),(87,122),(88,121),(89,120),(90,136),(91,135),(92,134),(93,133),(94,132),(95,131),(96,130),(97,129),(98,128),(99,127),(100,126),(101,125),(102,124)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 8A | 8B | 17A | ··· | 17H | 34A | ··· | 34H | 34I | ··· | 34X | 68A | ··· | 68H |
| order | 1 | 2 | 2 | 2 | 4 | 8 | 8 | 17 | ··· | 17 | 34 | ··· | 34 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 4 | 68 | 2 | 34 | 34 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D8 | D17 | D34 | C17⋊D4 | D4⋊D17 |
| kernel | D4⋊D17 | C17⋊3C8 | D68 | D4×C17 | C34 | C17 | D4 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 8 | 8 | 16 | 8 |
Matrix representation of D4⋊D17 ►in GL4(𝔽137) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 14 |
| 0 | 0 | 39 | 136 |
| 136 | 0 | 0 | 0 |
| 0 | 136 | 0 | 0 |
| 0 | 0 | 0 | 57 |
| 0 | 0 | 125 | 0 |
| 0 | 1 | 0 | 0 |
| 136 | 30 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 39 | 136 |
G:=sub<GL(4,GF(137))| [1,0,0,0,0,1,0,0,0,0,1,39,0,0,14,136],[136,0,0,0,0,136,0,0,0,0,0,125,0,0,57,0],[0,136,0,0,1,30,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,39,0,0,0,136] >;
D4⋊D17 in GAP, Magma, Sage, TeX
D_4\rtimes D_{17} % in TeX
G:=Group("D4:D17"); // GroupNames label
G:=SmallGroup(272,15);
// by ID
G=gap.SmallGroup(272,15);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,61,182,97,42,6404]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^17=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations
Export