direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D5×C17, C5⋊C34, C85⋊3C2, SmallGroup(170,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C17 |
Generators and relations for D5×C17
G = < a,b,c | a17=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C17
►On 85 points
Generators in S85
(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 30 65 80 49)(2 31 66 81 50)(3 32 67 82 51)(4 33 68 83 35)(5 34 52 84 36)(6 18 53 85 37)(7 19 54 69 38)(8 20 55 70 39)(9 21 56 71 40)(10 22 57 72 41)(11 23 58 73 42)(12 24 59 74 43)(13 25 60 75 44)(14 26 61 76 45)(15 27 62 77 46)(16 28 63 78 47)(17 29 64 79 48)
(1 49)(2 50)(3 51)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 48)(18 85)(19 69)(20 70)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 81)(32 82)(33 83)(34 84)
G:=sub<Sym(85)| (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), (1,30,65,80,49)(2,31,66,81,50)(3,32,67,82,51)(4,33,68,83,35)(5,34,52,84,36)(6,18,53,85,37)(7,19,54,69,38)(8,20,55,70,39)(9,21,56,71,40)(10,22,57,72,41)(11,23,58,73,42)(12,24,59,74,43)(13,25,60,75,44)(14,26,61,76,45)(15,27,62,77,46)(16,28,63,78,47)(17,29,64,79,48), (1,49)(2,50)(3,51)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,85)(19,69)(20,70)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,81)(32,82)(33,83)(34,84)>;
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), (1,30,65,80,49)(2,31,66,81,50)(3,32,67,82,51)(4,33,68,83,35)(5,34,52,84,36)(6,18,53,85,37)(7,19,54,69,38)(8,20,55,70,39)(9,21,56,71,40)(10,22,57,72,41)(11,23,58,73,42)(12,24,59,74,43)(13,25,60,75,44)(14,26,61,76,45)(15,27,62,77,46)(16,28,63,78,47)(17,29,64,79,48), (1,49)(2,50)(3,51)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,85)(19,69)(20,70)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,81)(32,82)(33,83)(34,84) );
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)], [(1,30,65,80,49),(2,31,66,81,50),(3,32,67,82,51),(4,33,68,83,35),(5,34,52,84,36),(6,18,53,85,37),(7,19,54,69,38),(8,20,55,70,39),(9,21,56,71,40),(10,22,57,72,41),(11,23,58,73,42),(12,24,59,74,43),(13,25,60,75,44),(14,26,61,76,45),(15,27,62,77,46),(16,28,63,78,47),(17,29,64,79,48)], [(1,49),(2,50),(3,51),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(17,48),(18,85),(19,69),(20,70),(21,71),(22,72),(23,73),(24,74),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,81),(32,82),(33,83),(34,84)]])
D5×C17 is a maximal subgroup of
C17⋊3F5
68 conjugacy classes
| class | 1 | 2 | 5A | 5B | 17A | ··· | 17P | 34A | ··· | 34P | 85A | ··· | 85AF |
| order | 1 | 2 | 5 | 5 | 17 | ··· | 17 | 34 | ··· | 34 | 85 | ··· | 85 |
| size | 1 | 5 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C17 | C34 | D5 | D5×C17 |
| kernel | D5×C17 | C85 | D5 | C5 | C17 | C1 |
| # reps | 1 | 1 | 16 | 16 | 2 | 32 |
Matrix representation of D5×C17 ►in GL2(𝔽1021) generated by
| 9 | 0 |
| 0 | 9 |
| 0 | 1 |
| 1020 | 563 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(1021))| [9,0,0,9],[0,1020,1,563],[0,1,1,0] >;
D5×C17 in GAP, Magma, Sage, TeX
D_5\times C_{17} % in TeX
G:=Group("D5xC17"); // GroupNames label
G:=SmallGroup(170,1);
// by ID
G=gap.SmallGroup(170,1);
# by ID
G:=PCGroup([3,-2,-17,-5,1226]);
// Polycyclic
G:=Group<a,b,c|a^17=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export