metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊2D17, C4.5D34, Dic34⋊3C2, C34.6C23, C68.5C22, C22.1D34, D34.2C22, Dic17.8C22, (C4×D17)⋊2C2, (D4×C17)⋊3C2, C17⋊2(C4○D4), C17⋊D4⋊2C2, (C2×C34).C22, (C2×Dic17)⋊3C2, C2.7(C22×D17), SmallGroup(272,41)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊2D17
G = < a,b,c,d | a4=b2=c17=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Smallest permutation representation of D4⋊2D17
►On 136 points
Generators in S136
(1 102 24 77)(2 86 25 78)(3 87 26 79)(4 88 27 80)(5 89 28 81)(6 90 29 82)(7 91 30 83)(8 92 31 84)(9 93 32 85)(10 94 33 69)(11 95 34 70)(12 96 18 71)(13 97 19 72)(14 98 20 73)(15 99 21 74)(16 100 22 75)(17 101 23 76)(35 126 62 106)(36 127 63 107)(37 128 64 108)(38 129 65 109)(39 130 66 110)(40 131 67 111)(41 132 68 112)(42 133 52 113)(43 134 53 114)(44 135 54 115)(45 136 55 116)(46 120 56 117)(47 121 57 118)(48 122 58 119)(49 123 59 103)(50 124 60 104)(51 125 61 105)
(1 103)(2 104)(3 105)(4 106)(5 107)(6 108)(7 109)(8 110)(9 111)(10 112)(11 113)(12 114)(13 115)(14 116)(15 117)(16 118)(17 119)(18 134)(19 135)(20 136)(21 120)(22 121)(23 122)(24 123)(25 124)(26 125)(27 126)(28 127)(29 128)(30 129)(31 130)(32 131)(33 132)(34 133)(35 80)(36 81)(37 82)(38 83)(39 84)(40 85)(41 69)(42 70)(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 77)(50 78)(51 79)(52 95)(53 96)(54 97)(55 98)(56 99)(57 100)(58 101)(59 102)(60 86)(61 87)(62 88)(63 89)(64 90)(65 91)(66 92)(67 93)(68 94)
(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 29)(19 28)(20 27)(21 26)(22 25)(23 24)(30 34)(31 33)(35 55)(36 54)(37 53)(38 52)(39 68)(40 67)(41 66)(42 65)(43 64)(44 63)(45 62)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)(69 84)(70 83)(71 82)(72 81)(73 80)(74 79)(75 78)(76 77)(86 100)(87 99)(88 98)(89 97)(90 96)(91 95)(92 94)(101 102)(103 122)(104 121)(105 120)(106 136)(107 135)(108 134)(109 133)(110 132)(111 131)(112 130)(113 129)(114 128)(115 127)(116 126)(117 125)(118 124)(119 123)
G:=sub<Sym(136)| (1,102,24,77)(2,86,25,78)(3,87,26,79)(4,88,27,80)(5,89,28,81)(6,90,29,82)(7,91,30,83)(8,92,31,84)(9,93,32,85)(10,94,33,69)(11,95,34,70)(12,96,18,71)(13,97,19,72)(14,98,20,73)(15,99,21,74)(16,100,22,75)(17,101,23,76)(35,126,62,106)(36,127,63,107)(37,128,64,108)(38,129,65,109)(39,130,66,110)(40,131,67,111)(41,132,68,112)(42,133,52,113)(43,134,53,114)(44,135,54,115)(45,136,55,116)(46,120,56,117)(47,121,57,118)(48,122,58,119)(49,123,59,103)(50,124,60,104)(51,125,61,105), (1,103)(2,104)(3,105)(4,106)(5,107)(6,108)(7,109)(8,110)(9,111)(10,112)(11,113)(12,114)(13,115)(14,116)(15,117)(16,118)(17,119)(18,134)(19,135)(20,136)(21,120)(22,121)(23,122)(24,123)(25,124)(26,125)(27,126)(28,127)(29,128)(30,129)(31,130)(32,131)(33,132)(34,133)(35,80)(36,81)(37,82)(38,83)(39,84)(40,85)(41,69)(42,70)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,79)(52,95)(53,96)(54,97)(55,98)(56,99)(57,100)(58,101)(59,102)(60,86)(61,87)(62,88)(63,89)(64,90)(65,91)(66,92)(67,93)(68,94), (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)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(30,34)(31,33)(35,55)(36,54)(37,53)(38,52)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(86,100)(87,99)(88,98)(89,97)(90,96)(91,95)(92,94)(101,102)(103,122)(104,121)(105,120)(106,136)(107,135)(108,134)(109,133)(110,132)(111,131)(112,130)(113,129)(114,128)(115,127)(116,126)(117,125)(118,124)(119,123)>;
G:=Group( (1,102,24,77)(2,86,25,78)(3,87,26,79)(4,88,27,80)(5,89,28,81)(6,90,29,82)(7,91,30,83)(8,92,31,84)(9,93,32,85)(10,94,33,69)(11,95,34,70)(12,96,18,71)(13,97,19,72)(14,98,20,73)(15,99,21,74)(16,100,22,75)(17,101,23,76)(35,126,62,106)(36,127,63,107)(37,128,64,108)(38,129,65,109)(39,130,66,110)(40,131,67,111)(41,132,68,112)(42,133,52,113)(43,134,53,114)(44,135,54,115)(45,136,55,116)(46,120,56,117)(47,121,57,118)(48,122,58,119)(49,123,59,103)(50,124,60,104)(51,125,61,105), (1,103)(2,104)(3,105)(4,106)(5,107)(6,108)(7,109)(8,110)(9,111)(10,112)(11,113)(12,114)(13,115)(14,116)(15,117)(16,118)(17,119)(18,134)(19,135)(20,136)(21,120)(22,121)(23,122)(24,123)(25,124)(26,125)(27,126)(28,127)(29,128)(30,129)(31,130)(32,131)(33,132)(34,133)(35,80)(36,81)(37,82)(38,83)(39,84)(40,85)(41,69)(42,70)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,79)(52,95)(53,96)(54,97)(55,98)(56,99)(57,100)(58,101)(59,102)(60,86)(61,87)(62,88)(63,89)(64,90)(65,91)(66,92)(67,93)(68,94), (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)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(30,34)(31,33)(35,55)(36,54)(37,53)(38,52)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(86,100)(87,99)(88,98)(89,97)(90,96)(91,95)(92,94)(101,102)(103,122)(104,121)(105,120)(106,136)(107,135)(108,134)(109,133)(110,132)(111,131)(112,130)(113,129)(114,128)(115,127)(116,126)(117,125)(118,124)(119,123) );
G=PermutationGroup([[(1,102,24,77),(2,86,25,78),(3,87,26,79),(4,88,27,80),(5,89,28,81),(6,90,29,82),(7,91,30,83),(8,92,31,84),(9,93,32,85),(10,94,33,69),(11,95,34,70),(12,96,18,71),(13,97,19,72),(14,98,20,73),(15,99,21,74),(16,100,22,75),(17,101,23,76),(35,126,62,106),(36,127,63,107),(37,128,64,108),(38,129,65,109),(39,130,66,110),(40,131,67,111),(41,132,68,112),(42,133,52,113),(43,134,53,114),(44,135,54,115),(45,136,55,116),(46,120,56,117),(47,121,57,118),(48,122,58,119),(49,123,59,103),(50,124,60,104),(51,125,61,105)], [(1,103),(2,104),(3,105),(4,106),(5,107),(6,108),(7,109),(8,110),(9,111),(10,112),(11,113),(12,114),(13,115),(14,116),(15,117),(16,118),(17,119),(18,134),(19,135),(20,136),(21,120),(22,121),(23,122),(24,123),(25,124),(26,125),(27,126),(28,127),(29,128),(30,129),(31,130),(32,131),(33,132),(34,133),(35,80),(36,81),(37,82),(38,83),(39,84),(40,85),(41,69),(42,70),(43,71),(44,72),(45,73),(46,74),(47,75),(48,76),(49,77),(50,78),(51,79),(52,95),(53,96),(54,97),(55,98),(56,99),(57,100),(58,101),(59,102),(60,86),(61,87),(62,88),(63,89),(64,90),(65,91),(66,92),(67,93),(68,94)], [(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),(18,29),(19,28),(20,27),(21,26),(22,25),(23,24),(30,34),(31,33),(35,55),(36,54),(37,53),(38,52),(39,68),(40,67),(41,66),(42,65),(43,64),(44,63),(45,62),(46,61),(47,60),(48,59),(49,58),(50,57),(51,56),(69,84),(70,83),(71,82),(72,81),(73,80),(74,79),(75,78),(76,77),(86,100),(87,99),(88,98),(89,97),(90,96),(91,95),(92,94),(101,102),(103,122),(104,121),(105,120),(106,136),(107,135),(108,134),(109,133),(110,132),(111,131),(112,130),(113,129),(114,128),(115,127),(116,126),(117,125),(118,124),(119,123)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 17A | ··· | 17H | 34A | ··· | 34H | 34I | ··· | 34X | 68A | ··· | 68H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 17 | ··· | 17 | 34 | ··· | 34 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 2 | 2 | 34 | 2 | 17 | 17 | 34 | 34 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4○D4 | D17 | D34 | D34 | D4⋊2D17 |
| kernel | D4⋊2D17 | Dic34 | C4×D17 | C2×Dic17 | C17⋊D4 | D4×C17 | C17 | D4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 8 | 8 | 16 | 8 |
Matrix representation of D4⋊2D17 ►in GL4(𝔽137) generated by
| 136 | 0 | 0 | 0 |
| 0 | 136 | 0 | 0 |
| 0 | 0 | 100 | 0 |
| 0 | 0 | 32 | 37 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 100 | 60 |
| 0 | 0 | 32 | 37 |
| 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 | 88 | 136 |
G:=sub<GL(4,GF(137))| [136,0,0,0,0,136,0,0,0,0,100,32,0,0,0,37],[1,0,0,0,0,1,0,0,0,0,100,32,0,0,60,37],[0,136,0,0,1,30,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,88,0,0,0,136] >;
D4⋊2D17 in GAP, Magma, Sage, TeX
D_4\rtimes_2D_{17} % in TeX
G:=Group("D4:2D17"); // GroupNames label
G:=SmallGroup(272,41);
// by ID
G=gap.SmallGroup(272,41);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,46,182,97,6404]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^17=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations
Export