direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C34, C23⋊C34, C68⋊4C22, C34.11C23, C4⋊(C2×C34), (C2×C68)⋊6C2, (C2×C4)⋊2C34, C22⋊(C2×C34), (C22×C34)⋊1C2, (C2×C34)⋊2C22, C2.1(C22×C34), SmallGroup(272,47)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C34
G = < a,b,c | a34=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 70 in 54 conjugacy classes, 38 normal (10 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, D4, C23, C2×D4, C17, C34, C34, C34, C68, C2×C34, C2×C34, C2×C34, C2×C68, D4×C17, C22×C34, D4×C34
Quotients: C1, C2, C22, D4, C23, C2×D4, C17, C34, C2×C34, D4×C17, C22×C34, D4×C34
►On 136 points
Generators in S136
(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)
(1 116 56 94)(2 117 57 95)(3 118 58 96)(4 119 59 97)(5 120 60 98)(6 121 61 99)(7 122 62 100)(8 123 63 101)(9 124 64 102)(10 125 65 69)(11 126 66 70)(12 127 67 71)(13 128 68 72)(14 129 35 73)(15 130 36 74)(16 131 37 75)(17 132 38 76)(18 133 39 77)(19 134 40 78)(20 135 41 79)(21 136 42 80)(22 103 43 81)(23 104 44 82)(24 105 45 83)(25 106 46 84)(26 107 47 85)(27 108 48 86)(28 109 49 87)(29 110 50 88)(30 111 51 89)(31 112 52 90)(32 113 53 91)(33 114 54 92)(34 115 55 93)
(1 133)(2 134)(3 135)(4 136)(5 103)(6 104)(7 105)(8 106)(9 107)(10 108)(11 109)(12 110)(13 111)(14 112)(15 113)(16 114)(17 115)(18 116)(19 117)(20 118)(21 119)(22 120)(23 121)(24 122)(25 123)(26 124)(27 125)(28 126)(29 127)(30 128)(31 129)(32 130)(33 131)(34 132)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 69)(49 70)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 78)(58 79)(59 80)(60 81)(61 82)(62 83)(63 84)(64 85)(65 86)(66 87)(67 88)(68 89)
G:=sub<Sym(136)| (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), (1,116,56,94)(2,117,57,95)(3,118,58,96)(4,119,59,97)(5,120,60,98)(6,121,61,99)(7,122,62,100)(8,123,63,101)(9,124,64,102)(10,125,65,69)(11,126,66,70)(12,127,67,71)(13,128,68,72)(14,129,35,73)(15,130,36,74)(16,131,37,75)(17,132,38,76)(18,133,39,77)(19,134,40,78)(20,135,41,79)(21,136,42,80)(22,103,43,81)(23,104,44,82)(24,105,45,83)(25,106,46,84)(26,107,47,85)(27,108,48,86)(28,109,49,87)(29,110,50,88)(30,111,51,89)(31,112,52,90)(32,113,53,91)(33,114,54,92)(34,115,55,93), (1,133)(2,134)(3,135)(4,136)(5,103)(6,104)(7,105)(8,106)(9,107)(10,108)(11,109)(12,110)(13,111)(14,112)(15,113)(16,114)(17,115)(18,116)(19,117)(20,118)(21,119)(22,120)(23,121)(24,122)(25,123)(26,124)(27,125)(28,126)(29,127)(30,128)(31,129)(32,130)(33,131)(34,132)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,69)(49,70)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(67,88)(68,89)>;
G:=Group( (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), (1,116,56,94)(2,117,57,95)(3,118,58,96)(4,119,59,97)(5,120,60,98)(6,121,61,99)(7,122,62,100)(8,123,63,101)(9,124,64,102)(10,125,65,69)(11,126,66,70)(12,127,67,71)(13,128,68,72)(14,129,35,73)(15,130,36,74)(16,131,37,75)(17,132,38,76)(18,133,39,77)(19,134,40,78)(20,135,41,79)(21,136,42,80)(22,103,43,81)(23,104,44,82)(24,105,45,83)(25,106,46,84)(26,107,47,85)(27,108,48,86)(28,109,49,87)(29,110,50,88)(30,111,51,89)(31,112,52,90)(32,113,53,91)(33,114,54,92)(34,115,55,93), (1,133)(2,134)(3,135)(4,136)(5,103)(6,104)(7,105)(8,106)(9,107)(10,108)(11,109)(12,110)(13,111)(14,112)(15,113)(16,114)(17,115)(18,116)(19,117)(20,118)(21,119)(22,120)(23,121)(24,122)(25,123)(26,124)(27,125)(28,126)(29,127)(30,128)(31,129)(32,130)(33,131)(34,132)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,69)(49,70)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(67,88)(68,89) );
G=PermutationGroup([[(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)], [(1,116,56,94),(2,117,57,95),(3,118,58,96),(4,119,59,97),(5,120,60,98),(6,121,61,99),(7,122,62,100),(8,123,63,101),(9,124,64,102),(10,125,65,69),(11,126,66,70),(12,127,67,71),(13,128,68,72),(14,129,35,73),(15,130,36,74),(16,131,37,75),(17,132,38,76),(18,133,39,77),(19,134,40,78),(20,135,41,79),(21,136,42,80),(22,103,43,81),(23,104,44,82),(24,105,45,83),(25,106,46,84),(26,107,47,85),(27,108,48,86),(28,109,49,87),(29,110,50,88),(30,111,51,89),(31,112,52,90),(32,113,53,91),(33,114,54,92),(34,115,55,93)], [(1,133),(2,134),(3,135),(4,136),(5,103),(6,104),(7,105),(8,106),(9,107),(10,108),(11,109),(12,110),(13,111),(14,112),(15,113),(16,114),(17,115),(18,116),(19,117),(20,118),(21,119),(22,120),(23,121),(24,122),(25,123),(26,124),(27,125),(28,126),(29,127),(30,128),(31,129),(32,130),(33,131),(34,132),(35,90),(36,91),(37,92),(38,93),(39,94),(40,95),(41,96),(42,97),(43,98),(44,99),(45,100),(46,101),(47,102),(48,69),(49,70),(50,71),(51,72),(52,73),(53,74),(54,75),(55,76),(56,77),(57,78),(58,79),(59,80),(60,81),(61,82),(62,83),(63,84),(64,85),(65,86),(66,87),(67,88),(68,89)]])
170 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 17A | ··· | 17P | 34A | ··· | 34AV | 34AW | ··· | 34DH | 68A | ··· | 68AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 17 | ··· | 17 | 34 | ··· | 34 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
170 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C17 | C34 | C34 | C34 | D4 | D4×C17 |
| kernel | D4×C34 | C2×C68 | D4×C17 | C22×C34 | C2×D4 | C2×C4 | D4 | C23 | C34 | C2 |
| # reps | 1 | 1 | 4 | 2 | 16 | 16 | 64 | 32 | 2 | 32 |
Matrix representation of D4×C34 ►in GL3(𝔽137) generated by
| 136 | 0 | 0 |
| 0 | 81 | 0 |
| 0 | 0 | 81 |
| 136 | 0 | 0 |
| 0 | 70 | 84 |
| 0 | 2 | 67 |
| 136 | 0 | 0 |
| 0 | 67 | 52 |
| 0 | 135 | 70 |
G:=sub<GL(3,GF(137))| [136,0,0,0,81,0,0,0,81],[136,0,0,0,70,2,0,84,67],[136,0,0,0,67,135,0,52,70] >;
D4×C34 in GAP, Magma, Sage, TeX
D_4\times C_{34} % in TeX
G:=Group("D4xC34"); // GroupNames label
G:=SmallGroup(272,47);
// by ID
G=gap.SmallGroup(272,47);
# by ID
G:=PCGroup([5,-2,-2,-2,-17,-2,1381]);
// Polycyclic
G:=Group<a,b,c|a^34=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations