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G = SD16×C17order 272 = 24·17

Direct product of C17 and SD16

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

Aliases: SD16×C17, Q8⋊C34, C82C34, D4.C34, C1366C2, C34.15D4, C68.18C22, C4.2(C2×C34), (Q8×C17)⋊4C2, C2.4(D4×C17), (D4×C17).2C2, SmallGroup(272,26)

Series: Derived Chief Lower central Upper central

C1C4 — SD16×C17
C1C2C4C68Q8×C17 — SD16×C17
C1C2C4 — SD16×C17
C1C34C68 — SD16×C17

Generators and relations for SD16×C17
 G = < a,b,c | a17=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

4C2
2C4
2C22
4C34
2C68
2C2×C34

Smallest permutation representation of SD16×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 54 107 18 81 89 126 38)(2 55 108 19 82 90 127 39)(3 56 109 20 83 91 128 40)(4 57 110 21 84 92 129 41)(5 58 111 22 85 93 130 42)(6 59 112 23 69 94 131 43)(7 60 113 24 70 95 132 44)(8 61 114 25 71 96 133 45)(9 62 115 26 72 97 134 46)(10 63 116 27 73 98 135 47)(11 64 117 28 74 99 136 48)(12 65 118 29 75 100 120 49)(13 66 119 30 76 101 121 50)(14 67 103 31 77 102 122 51)(15 68 104 32 78 86 123 35)(16 52 105 33 79 87 124 36)(17 53 106 34 80 88 125 37)
(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 52)(34 53)(35 86)(36 87)(37 88)(38 89)(39 90)(40 91)(41 92)(42 93)(43 94)(44 95)(45 96)(46 97)(47 98)(48 99)(49 100)(50 101)(51 102)(103 122)(104 123)(105 124)(106 125)(107 126)(108 127)(109 128)(110 129)(111 130)(112 131)(113 132)(114 133)(115 134)(116 135)(117 136)(118 120)(119 121)

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

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

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

119 conjugacy classes

class 1 2A2B4A4B8A8B17A···17P34A···34P34Q···34AF68A···68P68Q···68AF136A···136AF
order122448817···1734···3434···3468···6868···68136···136
size11424221···11···14···42···24···42···2

119 irreducible representations

dim111111112222
type+++++
imageC1C2C2C2C17C34C34C34D4SD16D4×C17SD16×C17
kernelSD16×C17C136D4×C17Q8×C17SD16C8D4Q8C34C17C2C1
# reps111116161616121632

Matrix representation of SD16×C17 in GL2(𝔽137) generated by

380
038
,
4394
4343
,
10
0136
G:=sub<GL(2,GF(137))| [38,0,0,38],[43,43,94,43],[1,0,0,136] >;

SD16×C17 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\times C_{17}
% in TeX

G:=Group("SD16xC17");
// GroupNames label

G:=SmallGroup(272,26);
// by ID

G=gap.SmallGroup(272,26);
# by ID

G:=PCGroup([5,-2,-2,-17,-2,-2,680,701,4083,2048,58]);
// 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^3>;
// generators/relations

Export

Subgroup lattice of SD16×C17 in TeX

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