direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×D17, C17⋊C10, C85⋊2C2, SmallGroup(170,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C5×D17 |
Generators and relations for C5×D17
G = < a,b,c | a5=b17=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C5×D17
►On 85 points
Generators in S85
(1 76 56 42 32)(2 77 57 43 33)(3 78 58 44 34)(4 79 59 45 18)(5 80 60 46 19)(6 81 61 47 20)(7 82 62 48 21)(8 83 63 49 22)(9 84 64 50 23)(10 85 65 51 24)(11 69 66 35 25)(12 70 67 36 26)(13 71 68 37 27)(14 72 52 38 28)(15 73 53 39 29)(16 74 54 40 30)(17 75 55 41 31)
(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 48)(36 47)(37 46)(38 45)(39 44)(40 43)(41 42)(49 51)(52 59)(53 58)(54 57)(55 56)(60 68)(61 67)(62 66)(63 65)(69 82)(70 81)(71 80)(72 79)(73 78)(74 77)(75 76)(83 85)
G:=sub<Sym(85)| (1,76,56,42,32)(2,77,57,43,33)(3,78,58,44,34)(4,79,59,45,18)(5,80,60,46,19)(6,81,61,47,20)(7,82,62,48,21)(8,83,63,49,22)(9,84,64,50,23)(10,85,65,51,24)(11,69,66,35,25)(12,70,67,36,26)(13,71,68,37,27)(14,72,52,38,28)(15,73,53,39,29)(16,74,54,40,30)(17,75,55,41,31), (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)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(49,51)(52,59)(53,58)(54,57)(55,56)(60,68)(61,67)(62,66)(63,65)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)(83,85)>;
G:=Group( (1,76,56,42,32)(2,77,57,43,33)(3,78,58,44,34)(4,79,59,45,18)(5,80,60,46,19)(6,81,61,47,20)(7,82,62,48,21)(8,83,63,49,22)(9,84,64,50,23)(10,85,65,51,24)(11,69,66,35,25)(12,70,67,36,26)(13,71,68,37,27)(14,72,52,38,28)(15,73,53,39,29)(16,74,54,40,30)(17,75,55,41,31), (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)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(49,51)(52,59)(53,58)(54,57)(55,56)(60,68)(61,67)(62,66)(63,65)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)(83,85) );
G=PermutationGroup([[(1,76,56,42,32),(2,77,57,43,33),(3,78,58,44,34),(4,79,59,45,18),(5,80,60,46,19),(6,81,61,47,20),(7,82,62,48,21),(8,83,63,49,22),(9,84,64,50,23),(10,85,65,51,24),(11,69,66,35,25),(12,70,67,36,26),(13,71,68,37,27),(14,72,52,38,28),(15,73,53,39,29),(16,74,54,40,30),(17,75,55,41,31)], [(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),(35,48),(36,47),(37,46),(38,45),(39,44),(40,43),(41,42),(49,51),(52,59),(53,58),(54,57),(55,56),(60,68),(61,67),(62,66),(63,65),(69,82),(70,81),(71,80),(72,79),(73,78),(74,77),(75,76),(83,85)]])
C5×D17 is a maximal subgroup of
C85⋊C4
50 conjugacy classes
| class | 1 | 2 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 17A | ··· | 17H | 85A | ··· | 85AF |
| order | 1 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 17 | ··· | 17 | 85 | ··· | 85 |
| size | 1 | 17 | 1 | 1 | 1 | 1 | 17 | 17 | 17 | 17 | 2 | ··· | 2 | 2 | ··· | 2 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C5 | C10 | D17 | C5×D17 |
| kernel | C5×D17 | C85 | D17 | C17 | C5 | C1 |
| # reps | 1 | 1 | 4 | 4 | 8 | 32 |
Matrix representation of C5×D17 ►in GL2(𝔽1021) generated by
| 676 | 0 |
| 0 | 676 |
| 0 | 1 |
| 1020 | 151 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(1021))| [676,0,0,676],[0,1020,1,151],[0,1,1,0] >;
C5×D17 in GAP, Magma, Sage, TeX
C_5\times D_{17} % in TeX
G:=Group("C5xD17"); // GroupNames label
G:=SmallGroup(170,2);
// by ID
G=gap.SmallGroup(170,2);
# by ID
G:=PCGroup([3,-2,-5,-17,1442]);
// Polycyclic
G:=Group<a,b,c|a^5=b^17=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export