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## G = D4×C34order 272 = 24·17

### Direct product of C34 and D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C34, C23⋊C34, C684C22, 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

 Derived series C1 — C2 — D4×C34
 Chief series C1 — C2 — C34 — C2×C34 — D4×C17 — D4×C34
 Lower central C1 — C2 — D4×C34
 Upper central C1 — C2×C34 — D4×C34

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

Smallest permutation representation of 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

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