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G = D8×C17order 272 = 24·17

Direct product of C17 and D8

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: D8×C17, D4⋊C34, C81C34, C1365C2, C34.14D4, C68.17C22, (D4×C17)⋊4C2, C4.1(C2×C34), C2.3(D4×C17), SmallGroup(272,25)

Series: Derived Chief Lower central Upper central

C1C4 — D8×C17
C1C2C4C68D4×C17 — D8×C17
C1C2C4 — D8×C17
C1C34C68 — D8×C17

Generators and relations for D8×C17
 G = < a,b,c | a17=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

4C2
4C2
2C22
2C22
4C34
4C34
2C2×C34
2C2×C34

Smallest permutation representation of D8×C17
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 94 35 134 116 59 20 79)(2 95 36 135 117 60 21 80)(3 96 37 136 118 61 22 81)(4 97 38 120 119 62 23 82)(5 98 39 121 103 63 24 83)(6 99 40 122 104 64 25 84)(7 100 41 123 105 65 26 85)(8 101 42 124 106 66 27 69)(9 102 43 125 107 67 28 70)(10 86 44 126 108 68 29 71)(11 87 45 127 109 52 30 72)(12 88 46 128 110 53 31 73)(13 89 47 129 111 54 32 74)(14 90 48 130 112 55 33 75)(15 91 49 131 113 56 34 76)(16 92 50 132 114 57 18 77)(17 93 51 133 115 58 19 78)
(1 79)(2 80)(3 81)(4 82)(5 83)(6 84)(7 85)(8 69)(9 70)(10 71)(11 72)(12 73)(13 74)(14 75)(15 76)(16 77)(17 78)(18 92)(19 93)(20 94)(21 95)(22 96)(23 97)(24 98)(25 99)(26 100)(27 101)(28 102)(29 86)(30 87)(31 88)(32 89)(33 90)(34 91)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(41 65)(42 66)(43 67)(44 68)(45 52)(46 53)(47 54)(48 55)(49 56)(50 57)(51 58)(103 121)(104 122)(105 123)(106 124)(107 125)(108 126)(109 127)(110 128)(111 129)(112 130)(113 131)(114 132)(115 133)(116 134)(117 135)(118 136)(119 120)

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), (1,94,35,134,116,59,20,79)(2,95,36,135,117,60,21,80)(3,96,37,136,118,61,22,81)(4,97,38,120,119,62,23,82)(5,98,39,121,103,63,24,83)(6,99,40,122,104,64,25,84)(7,100,41,123,105,65,26,85)(8,101,42,124,106,66,27,69)(9,102,43,125,107,67,28,70)(10,86,44,126,108,68,29,71)(11,87,45,127,109,52,30,72)(12,88,46,128,110,53,31,73)(13,89,47,129,111,54,32,74)(14,90,48,130,112,55,33,75)(15,91,49,131,113,56,34,76)(16,92,50,132,114,57,18,77)(17,93,51,133,115,58,19,78), (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,85)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,92)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,52)(46,53)(47,54)(48,55)(49,56)(50,57)(51,58)(103,121)(104,122)(105,123)(106,124)(107,125)(108,126)(109,127)(110,128)(111,129)(112,130)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,120)>;

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), (1,94,35,134,116,59,20,79)(2,95,36,135,117,60,21,80)(3,96,37,136,118,61,22,81)(4,97,38,120,119,62,23,82)(5,98,39,121,103,63,24,83)(6,99,40,122,104,64,25,84)(7,100,41,123,105,65,26,85)(8,101,42,124,106,66,27,69)(9,102,43,125,107,67,28,70)(10,86,44,126,108,68,29,71)(11,87,45,127,109,52,30,72)(12,88,46,128,110,53,31,73)(13,89,47,129,111,54,32,74)(14,90,48,130,112,55,33,75)(15,91,49,131,113,56,34,76)(16,92,50,132,114,57,18,77)(17,93,51,133,115,58,19,78), (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,85)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,92)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,52)(46,53)(47,54)(48,55)(49,56)(50,57)(51,58)(103,121)(104,122)(105,123)(106,124)(107,125)(108,126)(109,127)(110,128)(111,129)(112,130)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,120) );

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)], [(1,94,35,134,116,59,20,79),(2,95,36,135,117,60,21,80),(3,96,37,136,118,61,22,81),(4,97,38,120,119,62,23,82),(5,98,39,121,103,63,24,83),(6,99,40,122,104,64,25,84),(7,100,41,123,105,65,26,85),(8,101,42,124,106,66,27,69),(9,102,43,125,107,67,28,70),(10,86,44,126,108,68,29,71),(11,87,45,127,109,52,30,72),(12,88,46,128,110,53,31,73),(13,89,47,129,111,54,32,74),(14,90,48,130,112,55,33,75),(15,91,49,131,113,56,34,76),(16,92,50,132,114,57,18,77),(17,93,51,133,115,58,19,78)], [(1,79),(2,80),(3,81),(4,82),(5,83),(6,84),(7,85),(8,69),(9,70),(10,71),(11,72),(12,73),(13,74),(14,75),(15,76),(16,77),(17,78),(18,92),(19,93),(20,94),(21,95),(22,96),(23,97),(24,98),(25,99),(26,100),(27,101),(28,102),(29,86),(30,87),(31,88),(32,89),(33,90),(34,91),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64),(41,65),(42,66),(43,67),(44,68),(45,52),(46,53),(47,54),(48,55),(49,56),(50,57),(51,58),(103,121),(104,122),(105,123),(106,124),(107,125),(108,126),(109,127),(110,128),(111,129),(112,130),(113,131),(114,132),(115,133),(116,134),(117,135),(118,136),(119,120)])

119 conjugacy classes

class 1 2A2B2C 4 8A8B17A···17P34A···34P34Q···34AV68A···68P136A···136AF
order122248817···1734···3434···3468···68136···136
size11442221···11···14···42···22···2

119 irreducible representations

dim1111112222
type+++++
imageC1C2C2C17C34C34D4D8D4×C17D8×C17
kernelD8×C17C136D4×C17D8C8D4C34C17C2C1
# reps112161632121632

Matrix representation of D8×C17 in GL2(𝔽137) generated by

720
072
,
053
31106
,
10684
10631
G:=sub<GL(2,GF(137))| [72,0,0,72],[0,31,53,106],[106,106,84,31] >;

D8×C17 in GAP, Magma, Sage, TeX

D_8\times C_{17}
% in TeX

G:=Group("D8xC17");
// GroupNames label

G:=SmallGroup(272,25);
// by ID

G=gap.SmallGroup(272,25);
# by ID

G:=PCGroup([5,-2,-2,-17,-2,-2,701,4083,2048,58]);
// Polycyclic

G:=Group<a,b,c|a^17=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D8×C17 in TeX

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