direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C17×F5, C5⋊C68, C85⋊6C4, D5.C34, (D5×C17).2C2, SmallGroup(340,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C17×F5 |
Generators and relations for C17×F5
G = < a,b,c | a17=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of C17×F5
►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 62 81 19 46)(2 63 82 20 47)(3 64 83 21 48)(4 65 84 22 49)(5 66 85 23 50)(6 67 69 24 51)(7 68 70 25 35)(8 52 71 26 36)(9 53 72 27 37)(10 54 73 28 38)(11 55 74 29 39)(12 56 75 30 40)(13 57 76 31 41)(14 58 77 32 42)(15 59 78 33 43)(16 60 79 34 44)(17 61 80 18 45)
(18 61 80 45)(19 62 81 46)(20 63 82 47)(21 64 83 48)(22 65 84 49)(23 66 85 50)(24 67 69 51)(25 68 70 35)(26 52 71 36)(27 53 72 37)(28 54 73 38)(29 55 74 39)(30 56 75 40)(31 57 76 41)(32 58 77 42)(33 59 78 43)(34 60 79 44)
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), (1,62,81,19,46)(2,63,82,20,47)(3,64,83,21,48)(4,65,84,22,49)(5,66,85,23,50)(6,67,69,24,51)(7,68,70,25,35)(8,52,71,26,36)(9,53,72,27,37)(10,54,73,28,38)(11,55,74,29,39)(12,56,75,30,40)(13,57,76,31,41)(14,58,77,32,42)(15,59,78,33,43)(16,60,79,34,44)(17,61,80,18,45), (18,61,80,45)(19,62,81,46)(20,63,82,47)(21,64,83,48)(22,65,84,49)(23,66,85,50)(24,67,69,51)(25,68,70,35)(26,52,71,36)(27,53,72,37)(28,54,73,38)(29,55,74,39)(30,56,75,40)(31,57,76,41)(32,58,77,42)(33,59,78,43)(34,60,79,44)>;
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), (1,62,81,19,46)(2,63,82,20,47)(3,64,83,21,48)(4,65,84,22,49)(5,66,85,23,50)(6,67,69,24,51)(7,68,70,25,35)(8,52,71,26,36)(9,53,72,27,37)(10,54,73,28,38)(11,55,74,29,39)(12,56,75,30,40)(13,57,76,31,41)(14,58,77,32,42)(15,59,78,33,43)(16,60,79,34,44)(17,61,80,18,45), (18,61,80,45)(19,62,81,46)(20,63,82,47)(21,64,83,48)(22,65,84,49)(23,66,85,50)(24,67,69,51)(25,68,70,35)(26,52,71,36)(27,53,72,37)(28,54,73,38)(29,55,74,39)(30,56,75,40)(31,57,76,41)(32,58,77,42)(33,59,78,43)(34,60,79,44) );
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)], [(1,62,81,19,46),(2,63,82,20,47),(3,64,83,21,48),(4,65,84,22,49),(5,66,85,23,50),(6,67,69,24,51),(7,68,70,25,35),(8,52,71,26,36),(9,53,72,27,37),(10,54,73,28,38),(11,55,74,29,39),(12,56,75,30,40),(13,57,76,31,41),(14,58,77,32,42),(15,59,78,33,43),(16,60,79,34,44),(17,61,80,18,45)], [(18,61,80,45),(19,62,81,46),(20,63,82,47),(21,64,83,48),(22,65,84,49),(23,66,85,50),(24,67,69,51),(25,68,70,35),(26,52,71,36),(27,53,72,37),(28,54,73,38),(29,55,74,39),(30,56,75,40),(31,57,76,41),(32,58,77,42),(33,59,78,43),(34,60,79,44)]])
85 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5 | 17A | ··· | 17P | 34A | ··· | 34P | 68A | ··· | 68AF | 85A | ··· | 85P |
| order | 1 | 2 | 4 | 4 | 5 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 | 85 | ··· | 85 |
| size | 1 | 5 | 5 | 5 | 4 | 1 | ··· | 1 | 5 | ··· | 5 | 5 | ··· | 5 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | |||||
| image | C1 | C2 | C4 | C17 | C34 | C68 | F5 | C17×F5 |
| kernel | C17×F5 | D5×C17 | C85 | F5 | D5 | C5 | C17 | C1 |
| # reps | 1 | 1 | 2 | 16 | 16 | 32 | 1 | 16 |
Matrix representation of C17×F5 ►in GL4(𝔽1021) generated by
| 507 | 0 | 0 | 0 |
| 0 | 507 | 0 | 0 |
| 0 | 0 | 507 | 0 |
| 0 | 0 | 0 | 507 |
| 1020 | 1020 | 1020 | 1020 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1020 | 1020 | 1020 | 1020 |
G:=sub<GL(4,GF(1021))| [507,0,0,0,0,507,0,0,0,0,507,0,0,0,0,507],[1020,1,0,0,1020,0,1,0,1020,0,0,1,1020,0,0,0],[1,0,0,1020,0,0,1,1020,0,0,0,1020,0,1,0,1020] >;
C17×F5 in GAP, Magma, Sage, TeX
C_{17}\times F_5 % in TeX
G:=Group("C17xF5"); // GroupNames label
G:=SmallGroup(340,7);
// by ID
G=gap.SmallGroup(340,7);
# by ID
G:=PCGroup([4,-2,-17,-2,-5,136,2179,139]);
// Polycyclic
G:=Group<a,b,c|a^17=b^5=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
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