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## G = D5×D17order 340 = 22·5·17

### Direct product of D5 and D17

Aliases: D5×D17, D85⋊C2, C51D34, C85⋊C22, C171D10, (D5×C17)⋊C2, (C5×D17)⋊C2, SmallGroup(340,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C85 — D5×D17
 Chief series C1 — C17 — C85 — C5×D17 — D5×D17
 Lower central C85 — D5×D17
 Upper central C1

Generators and relations for D5×D17
G = < a,b,c,d | a5=b2=c17=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

5C2
17C2
85C2
85C22
17C10
17D5
5C34
5D17
17D10
5D34

Smallest permutation representation of D5×D17
On 85 points
Generators in S85
(1 77 61 40 32)(2 78 62 41 33)(3 79 63 42 34)(4 80 64 43 18)(5 81 65 44 19)(6 82 66 45 20)(7 83 67 46 21)(8 84 68 47 22)(9 85 52 48 23)(10 69 53 49 24)(11 70 54 50 25)(12 71 55 51 26)(13 72 56 35 27)(14 73 57 36 28)(15 74 58 37 29)(16 75 59 38 30)(17 76 60 39 31)
(1 32)(2 33)(3 34)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 31)(35 72)(36 73)(37 74)(38 75)(39 76)(40 77)(41 78)(42 79)(43 80)(44 81)(45 82)(46 83)(47 84)(48 85)(49 69)(50 70)(51 71)
(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)
(1 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(18 28)(19 27)(20 26)(21 25)(22 24)(29 34)(30 33)(31 32)(35 44)(36 43)(37 42)(38 41)(39 40)(45 51)(46 50)(47 49)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)(69 84)(70 83)(71 82)(72 81)(73 80)(74 79)(75 78)(76 77)

G:=sub<Sym(85)| (1,77,61,40,32)(2,78,62,41,33)(3,79,63,42,34)(4,80,64,43,18)(5,81,65,44,19)(6,82,66,45,20)(7,83,67,46,21)(8,84,68,47,22)(9,85,52,48,23)(10,69,53,49,24)(11,70,54,50,25)(12,71,55,51,26)(13,72,56,35,27)(14,73,57,36,28)(15,74,58,37,29)(16,75,59,38,30)(17,76,60,39,31), (1,32)(2,33)(3,34)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,31)(35,72)(36,73)(37,74)(38,75)(39,76)(40,77)(41,78)(42,79)(43,80)(44,81)(45,82)(46,83)(47,84)(48,85)(49,69)(50,70)(51,71), (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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,28)(19,27)(20,26)(21,25)(22,24)(29,34)(30,33)(31,32)(35,44)(36,43)(37,42)(38,41)(39,40)(45,51)(46,50)(47,49)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)>;

G:=Group( (1,77,61,40,32)(2,78,62,41,33)(3,79,63,42,34)(4,80,64,43,18)(5,81,65,44,19)(6,82,66,45,20)(7,83,67,46,21)(8,84,68,47,22)(9,85,52,48,23)(10,69,53,49,24)(11,70,54,50,25)(12,71,55,51,26)(13,72,56,35,27)(14,73,57,36,28)(15,74,58,37,29)(16,75,59,38,30)(17,76,60,39,31), (1,32)(2,33)(3,34)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,31)(35,72)(36,73)(37,74)(38,75)(39,76)(40,77)(41,78)(42,79)(43,80)(44,81)(45,82)(46,83)(47,84)(48,85)(49,69)(50,70)(51,71), (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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,28)(19,27)(20,26)(21,25)(22,24)(29,34)(30,33)(31,32)(35,44)(36,43)(37,42)(38,41)(39,40)(45,51)(46,50)(47,49)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77) );

G=PermutationGroup([[(1,77,61,40,32),(2,78,62,41,33),(3,79,63,42,34),(4,80,64,43,18),(5,81,65,44,19),(6,82,66,45,20),(7,83,67,46,21),(8,84,68,47,22),(9,85,52,48,23),(10,69,53,49,24),(11,70,54,50,25),(12,71,55,51,26),(13,72,56,35,27),(14,73,57,36,28),(15,74,58,37,29),(16,75,59,38,30),(17,76,60,39,31)], [(1,32),(2,33),(3,34),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(15,29),(16,30),(17,31),(35,72),(36,73),(37,74),(38,75),(39,76),(40,77),(41,78),(42,79),(43,80),(44,81),(45,82),(46,83),(47,84),(48,85),(49,69),(50,70),(51,71)], [(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)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(18,28),(19,27),(20,26),(21,25),(22,24),(29,34),(30,33),(31,32),(35,44),(36,43),(37,42),(38,41),(39,40),(45,51),(46,50),(47,49),(53,68),(54,67),(55,66),(56,65),(57,64),(58,63),(59,62),(60,61),(69,84),(70,83),(71,82),(72,81),(73,80),(74,79),(75,78),(76,77)]])

40 conjugacy classes

 class 1 2A 2B 2C 5A 5B 10A 10B 17A ··· 17H 34A ··· 34H 85A ··· 85P order 1 2 2 2 5 5 10 10 17 ··· 17 34 ··· 34 85 ··· 85 size 1 5 17 85 2 2 34 34 2 ··· 2 10 ··· 10 4 ··· 4

40 irreducible representations

 dim 1 1 1 1 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 D5 D10 D17 D34 D5×D17 kernel D5×D17 D5×C17 C5×D17 D85 D17 C17 D5 C5 C1 # reps 1 1 1 1 2 2 8 8 16

Matrix representation of D5×D17 in GL4(𝔽1021) generated by

 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1020 457
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
,
 0 1 0 0 1020 578 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(1021))| [1,0,0,0,0,1,0,0,0,0,0,1020,0,0,1,457],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[0,1020,0,0,1,578,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;

D5×D17 in GAP, Magma, Sage, TeX

D_5\times D_{17}
% in TeX

G:=Group("D5xD17");
// GroupNames label

G:=SmallGroup(340,11);
// by ID

G=gap.SmallGroup(340,11);
# by ID

G:=PCGroup([4,-2,-2,-5,-17,102,5123]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^17=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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