metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.D17, C34.8D4, C4.2D34, C17⋊2SD16, Dic34⋊2C2, C68.2C22, C17⋊3C8⋊2C2, (D4×C17).1C2, C2.5(C17⋊D4), SmallGroup(272,16)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.D17
G = < a,b,c,d | a4=b2=c17=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >
Smallest permutation representation of D4.D17
►On 136 points
Generators in S136
(1 64 30 35)(2 65 31 36)(3 66 32 37)(4 67 33 38)(5 68 34 39)(6 52 18 40)(7 53 19 41)(8 54 20 42)(9 55 21 43)(10 56 22 44)(11 57 23 45)(12 58 24 46)(13 59 25 47)(14 60 26 48)(15 61 27 49)(16 62 28 50)(17 63 29 51)(69 103 86 120)(70 104 87 121)(71 105 88 122)(72 106 89 123)(73 107 90 124)(74 108 91 125)(75 109 92 126)(76 110 93 127)(77 111 94 128)(78 112 95 129)(79 113 96 130)(80 114 97 131)(81 115 98 132)(82 116 99 133)(83 117 100 134)(84 118 101 135)(85 119 102 136)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 49)(16 50)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 57)(24 58)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 65)(32 66)(33 67)(34 68)(69 86)(70 87)(71 88)(72 89)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 97)(81 98)(82 99)(83 100)(84 101)(85 102)
(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)
(1 102 30 85)(2 101 31 84)(3 100 32 83)(4 99 33 82)(5 98 34 81)(6 97 18 80)(7 96 19 79)(8 95 20 78)(9 94 21 77)(10 93 22 76)(11 92 23 75)(12 91 24 74)(13 90 25 73)(14 89 26 72)(15 88 27 71)(16 87 28 70)(17 86 29 69)(35 136 64 119)(36 135 65 118)(37 134 66 117)(38 133 67 116)(39 132 68 115)(40 131 52 114)(41 130 53 113)(42 129 54 112)(43 128 55 111)(44 127 56 110)(45 126 57 109)(46 125 58 108)(47 124 59 107)(48 123 60 106)(49 122 61 105)(50 121 62 104)(51 120 63 103)
G:=sub<Sym(136)| (1,64,30,35)(2,65,31,36)(3,66,32,37)(4,67,33,38)(5,68,34,39)(6,52,18,40)(7,53,19,41)(8,54,20,42)(9,55,21,43)(10,56,22,44)(11,57,23,45)(12,58,24,46)(13,59,25,47)(14,60,26,48)(15,61,27,49)(16,62,28,50)(17,63,29,51)(69,103,86,120)(70,104,87,121)(71,105,88,122)(72,106,89,123)(73,107,90,124)(74,108,91,125)(75,109,92,126)(76,110,93,127)(77,111,94,128)(78,112,95,129)(79,113,96,130)(80,114,97,131)(81,115,98,132)(82,116,99,133)(83,117,100,134)(84,118,101,135)(85,119,102,136), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,49)(16,50)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68)(69,86)(70,87)(71,88)(72,89)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,97)(81,98)(82,99)(83,100)(84,101)(85,102), (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), (1,102,30,85)(2,101,31,84)(3,100,32,83)(4,99,33,82)(5,98,34,81)(6,97,18,80)(7,96,19,79)(8,95,20,78)(9,94,21,77)(10,93,22,76)(11,92,23,75)(12,91,24,74)(13,90,25,73)(14,89,26,72)(15,88,27,71)(16,87,28,70)(17,86,29,69)(35,136,64,119)(36,135,65,118)(37,134,66,117)(38,133,67,116)(39,132,68,115)(40,131,52,114)(41,130,53,113)(42,129,54,112)(43,128,55,111)(44,127,56,110)(45,126,57,109)(46,125,58,108)(47,124,59,107)(48,123,60,106)(49,122,61,105)(50,121,62,104)(51,120,63,103)>;
G:=Group( (1,64,30,35)(2,65,31,36)(3,66,32,37)(4,67,33,38)(5,68,34,39)(6,52,18,40)(7,53,19,41)(8,54,20,42)(9,55,21,43)(10,56,22,44)(11,57,23,45)(12,58,24,46)(13,59,25,47)(14,60,26,48)(15,61,27,49)(16,62,28,50)(17,63,29,51)(69,103,86,120)(70,104,87,121)(71,105,88,122)(72,106,89,123)(73,107,90,124)(74,108,91,125)(75,109,92,126)(76,110,93,127)(77,111,94,128)(78,112,95,129)(79,113,96,130)(80,114,97,131)(81,115,98,132)(82,116,99,133)(83,117,100,134)(84,118,101,135)(85,119,102,136), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,49)(16,50)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68)(69,86)(70,87)(71,88)(72,89)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,97)(81,98)(82,99)(83,100)(84,101)(85,102), (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), (1,102,30,85)(2,101,31,84)(3,100,32,83)(4,99,33,82)(5,98,34,81)(6,97,18,80)(7,96,19,79)(8,95,20,78)(9,94,21,77)(10,93,22,76)(11,92,23,75)(12,91,24,74)(13,90,25,73)(14,89,26,72)(15,88,27,71)(16,87,28,70)(17,86,29,69)(35,136,64,119)(36,135,65,118)(37,134,66,117)(38,133,67,116)(39,132,68,115)(40,131,52,114)(41,130,53,113)(42,129,54,112)(43,128,55,111)(44,127,56,110)(45,126,57,109)(46,125,58,108)(47,124,59,107)(48,123,60,106)(49,122,61,105)(50,121,62,104)(51,120,63,103) );
G=PermutationGroup([[(1,64,30,35),(2,65,31,36),(3,66,32,37),(4,67,33,38),(5,68,34,39),(6,52,18,40),(7,53,19,41),(8,54,20,42),(9,55,21,43),(10,56,22,44),(11,57,23,45),(12,58,24,46),(13,59,25,47),(14,60,26,48),(15,61,27,49),(16,62,28,50),(17,63,29,51),(69,103,86,120),(70,104,87,121),(71,105,88,122),(72,106,89,123),(73,107,90,124),(74,108,91,125),(75,109,92,126),(76,110,93,127),(77,111,94,128),(78,112,95,129),(79,113,96,130),(80,114,97,131),(81,115,98,132),(82,116,99,133),(83,117,100,134),(84,118,101,135),(85,119,102,136)], [(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,41),(8,42),(9,43),(10,44),(11,45),(12,46),(13,47),(14,48),(15,49),(16,50),(17,51),(18,52),(19,53),(20,54),(21,55),(22,56),(23,57),(24,58),(25,59),(26,60),(27,61),(28,62),(29,63),(30,64),(31,65),(32,66),(33,67),(34,68),(69,86),(70,87),(71,88),(72,89),(73,90),(74,91),(75,92),(76,93),(77,94),(78,95),(79,96),(80,97),(81,98),(82,99),(83,100),(84,101),(85,102)], [(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)], [(1,102,30,85),(2,101,31,84),(3,100,32,83),(4,99,33,82),(5,98,34,81),(6,97,18,80),(7,96,19,79),(8,95,20,78),(9,94,21,77),(10,93,22,76),(11,92,23,75),(12,91,24,74),(13,90,25,73),(14,89,26,72),(15,88,27,71),(16,87,28,70),(17,86,29,69),(35,136,64,119),(36,135,65,118),(37,134,66,117),(38,133,67,116),(39,132,68,115),(40,131,52,114),(41,130,53,113),(42,129,54,112),(43,128,55,111),(44,127,56,110),(45,126,57,109),(46,125,58,108),(47,124,59,107),(48,123,60,106),(49,122,61,105),(50,121,62,104),(51,120,63,103)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 17A | ··· | 17H | 34A | ··· | 34H | 34I | ··· | 34X | 68A | ··· | 68H |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 17 | ··· | 17 | 34 | ··· | 34 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 4 | 2 | 68 | 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 | SD16 | D17 | D34 | C17⋊D4 | D4.D17 |
| kernel | D4.D17 | C17⋊3C8 | Dic34 | 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 | 119 |
| 0 | 0 | 61 | 136 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 119 |
| 0 | 0 | 0 | 136 |
| 103 | 1 | 0 | 0 |
| 22 | 92 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 71 | 76 | 0 | 0 |
| 13 | 66 | 0 | 0 |
| 0 | 0 | 0 | 89 |
| 0 | 0 | 20 | 0 |
G:=sub<GL(4,GF(137))| [1,0,0,0,0,1,0,0,0,0,1,61,0,0,119,136],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,119,136],[103,22,0,0,1,92,0,0,0,0,1,0,0,0,0,1],[71,13,0,0,76,66,0,0,0,0,0,20,0,0,89,0] >;
D4.D17 in GAP, Magma, Sage, TeX
D_4.D_{17} % in TeX
G:=Group("D4.D17"); // GroupNames label
G:=SmallGroup(272,16);
// by ID
G=gap.SmallGroup(272,16);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,40,61,182,97,42,6404]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^17=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations
Export