metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C85⋊4C4, C5⋊Dic17, C17⋊3F5, D5.D17, (D5×C17).1C2, SmallGroup(340,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C85 — C17⋊3F5 |
Generators and relations for C17⋊3F5
G = < a,b,c | a17=b5=c4=1, ab=ba, cac-1=a-1, cbc-1=b3 >
Smallest permutation representation of C17⋊3F5
►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 27 47 58 81)(2 28 48 59 82)(3 29 49 60 83)(4 30 50 61 84)(5 31 51 62 85)(6 32 35 63 69)(7 33 36 64 70)(8 34 37 65 71)(9 18 38 66 72)(10 19 39 67 73)(11 20 40 68 74)(12 21 41 52 75)(13 22 42 53 76)(14 23 43 54 77)(15 24 44 55 78)(16 25 45 56 79)(17 26 46 57 80)
(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)(18 39 72 67)(19 38 73 66)(20 37 74 65)(21 36 75 64)(22 35 76 63)(23 51 77 62)(24 50 78 61)(25 49 79 60)(26 48 80 59)(27 47 81 58)(28 46 82 57)(29 45 83 56)(30 44 84 55)(31 43 85 54)(32 42 69 53)(33 41 70 52)(34 40 71 68)
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), (1,27,47,58,81)(2,28,48,59,82)(3,29,49,60,83)(4,30,50,61,84)(5,31,51,62,85)(6,32,35,63,69)(7,33,36,64,70)(8,34,37,65,71)(9,18,38,66,72)(10,19,39,67,73)(11,20,40,68,74)(12,21,41,52,75)(13,22,42,53,76)(14,23,43,54,77)(15,24,44,55,78)(16,25,45,56,79)(17,26,46,57,80), (2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(18,39,72,67)(19,38,73,66)(20,37,74,65)(21,36,75,64)(22,35,76,63)(23,51,77,62)(24,50,78,61)(25,49,79,60)(26,48,80,59)(27,47,81,58)(28,46,82,57)(29,45,83,56)(30,44,84,55)(31,43,85,54)(32,42,69,53)(33,41,70,52)(34,40,71,68)>;
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), (1,27,47,58,81)(2,28,48,59,82)(3,29,49,60,83)(4,30,50,61,84)(5,31,51,62,85)(6,32,35,63,69)(7,33,36,64,70)(8,34,37,65,71)(9,18,38,66,72)(10,19,39,67,73)(11,20,40,68,74)(12,21,41,52,75)(13,22,42,53,76)(14,23,43,54,77)(15,24,44,55,78)(16,25,45,56,79)(17,26,46,57,80), (2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(18,39,72,67)(19,38,73,66)(20,37,74,65)(21,36,75,64)(22,35,76,63)(23,51,77,62)(24,50,78,61)(25,49,79,60)(26,48,80,59)(27,47,81,58)(28,46,82,57)(29,45,83,56)(30,44,84,55)(31,43,85,54)(32,42,69,53)(33,41,70,52)(34,40,71,68) );
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)], [(1,27,47,58,81),(2,28,48,59,82),(3,29,49,60,83),(4,30,50,61,84),(5,31,51,62,85),(6,32,35,63,69),(7,33,36,64,70),(8,34,37,65,71),(9,18,38,66,72),(10,19,39,67,73),(11,20,40,68,74),(12,21,41,52,75),(13,22,42,53,76),(14,23,43,54,77),(15,24,44,55,78),(16,25,45,56,79),(17,26,46,57,80)], [(2,17),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10),(18,39,72,67),(19,38,73,66),(20,37,74,65),(21,36,75,64),(22,35,76,63),(23,51,77,62),(24,50,78,61),(25,49,79,60),(26,48,80,59),(27,47,81,58),(28,46,82,57),(29,45,83,56),(30,44,84,55),(31,43,85,54),(32,42,69,53),(33,41,70,52),(34,40,71,68)]])
37 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5 | 17A | ··· | 17H | 34A | ··· | 34H | 85A | ··· | 85P |
| order | 1 | 2 | 4 | 4 | 5 | 17 | ··· | 17 | 34 | ··· | 34 | 85 | ··· | 85 |
| size | 1 | 5 | 85 | 85 | 4 | 2 | ··· | 2 | 10 | ··· | 10 | 4 | ··· | 4 |
37 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | - | + | ||
| image | C1 | C2 | C4 | D17 | Dic17 | F5 | C17⋊3F5 |
| kernel | C17⋊3F5 | D5×C17 | C85 | D5 | C5 | C17 | C1 |
| # reps | 1 | 1 | 2 | 8 | 8 | 1 | 16 |
Matrix representation of C17⋊3F5 ►in GL4(𝔽1021) generated by
| 0 | 1 | 0 | 0 |
| 1020 | 278 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1020 | 278 |
| 843 | 644 | 1020 | 0 |
| 377 | 179 | 0 | 1020 |
| 844 | 644 | 1020 | 0 |
| 377 | 180 | 0 | 1020 |
| 178 | 377 | 843 | 644 |
| 853 | 843 | 168 | 178 |
| 1 | 0 | 0 | 0 |
| 278 | 1020 | 0 | 0 |
G:=sub<GL(4,GF(1021))| [0,1020,0,0,1,278,0,0,0,0,0,1020,0,0,1,278],[843,377,844,377,644,179,644,180,1020,0,1020,0,0,1020,0,1020],[178,853,1,278,377,843,0,1020,843,168,0,0,644,178,0,0] >;
C17⋊3F5 in GAP, Magma, Sage, TeX
C_{17}\rtimes_3F_5 % in TeX
G:=Group("C17:3F5"); // GroupNames label
G:=SmallGroup(340,8);
// by ID
G=gap.SmallGroup(340,8);
# by ID
G:=PCGroup([4,-2,-2,-5,-17,8,146,102,5123]);
// Polycyclic
G:=Group<a,b,c|a^17=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
// generators/relations
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