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G = D5xD17order 340 = 22·5·17

Direct product of D5 and D17

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D5xD17, D85:C2, C5:1D34, C85:C22, C17:1D10, (D5xC17):C2, (C5xD17):C2, SmallGroup(340,11)

Series: Derived Chief Lower central Upper central

C1C85 — D5xD17
C1C17C85C5xD17 — D5xD17
C85 — D5xD17
C1

Generators and relations for D5xD17
 G = < a,b,c,d | a5=b2=c17=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 268 in 20 conjugacy classes, 10 normal (all characteristic)
Quotients: C1, C2, C22, D5, D10, D17, D34, D5xD17
5C2
17C2
85C2
85C22
17C10
17D5
5C34
5D17
17D10
5D34

Smallest permutation representation of D5xD17
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 2A2B2C5A5B10A10B17A···17H34A···34H85A···85P
order122255101017···1734···3485···85
size1517852234342···210···104···4

40 irreducible representations

dim111122224
type+++++++++
imageC1C2C2C2D5D10D17D34D5xD17
kernelD5xD17D5xC17C5xD17D85D17C17D5C5C1
# reps1111228816

Matrix representation of D5xD17 in GL4(F1021) generated by

1000
0100
0001
001020457
,
1000
0100
0001
0010
,
0100
102057800
0010
0001
,
0100
1000
0010
0001
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] >;

D5xD17 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

Export

Subgroup lattice of D5xD17 in TeX

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