direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×D17, D85⋊C2, C5⋊1D34, C85⋊C22, C17⋊1D10, (D5×C17)⋊C2, (C5×D17)⋊C2, SmallGroup(340,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C85 — D5×D17 |
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 >
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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