direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×C17⋊C4, C17⋊C20, C85⋊5C4, D17.C10, (C5×D17).2C2, SmallGroup(340,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C5×C17⋊C4 |
Generators and relations for C5×C17⋊C4
G = < a,b,c | a5=b17=c4=1, ab=ba, ac=ca, cbc-1=b4 >
Smallest permutation representation of C5×C17⋊C4
►On 85 points
Generators in S85
(1 69 52 35 18)(2 70 53 36 19)(3 71 54 37 20)(4 72 55 38 21)(5 73 56 39 22)(6 74 57 40 23)(7 75 58 41 24)(8 76 59 42 25)(9 77 60 43 26)(10 78 61 44 27)(11 79 62 45 28)(12 80 63 46 29)(13 81 64 47 30)(14 82 65 48 31)(15 83 66 49 32)(16 84 67 50 33)(17 85 68 51 34)
(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)
(2 14 17 5)(3 10 16 9)(4 6 15 13)(7 11 12 8)(19 31 34 22)(20 27 33 26)(21 23 32 30)(24 28 29 25)(36 48 51 39)(37 44 50 43)(38 40 49 47)(41 45 46 42)(53 65 68 56)(54 61 67 60)(55 57 66 64)(58 62 63 59)(70 82 85 73)(71 78 84 77)(72 74 83 81)(75 79 80 76)
G:=sub<Sym(85)| (1,69,52,35,18)(2,70,53,36,19)(3,71,54,37,20)(4,72,55,38,21)(5,73,56,39,22)(6,74,57,40,23)(7,75,58,41,24)(8,76,59,42,25)(9,77,60,43,26)(10,78,61,44,27)(11,79,62,45,28)(12,80,63,46,29)(13,81,64,47,30)(14,82,65,48,31)(15,83,66,49,32)(16,84,67,50,33)(17,85,68,51,34), (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), (2,14,17,5)(3,10,16,9)(4,6,15,13)(7,11,12,8)(19,31,34,22)(20,27,33,26)(21,23,32,30)(24,28,29,25)(36,48,51,39)(37,44,50,43)(38,40,49,47)(41,45,46,42)(53,65,68,56)(54,61,67,60)(55,57,66,64)(58,62,63,59)(70,82,85,73)(71,78,84,77)(72,74,83,81)(75,79,80,76)>;
G:=Group( (1,69,52,35,18)(2,70,53,36,19)(3,71,54,37,20)(4,72,55,38,21)(5,73,56,39,22)(6,74,57,40,23)(7,75,58,41,24)(8,76,59,42,25)(9,77,60,43,26)(10,78,61,44,27)(11,79,62,45,28)(12,80,63,46,29)(13,81,64,47,30)(14,82,65,48,31)(15,83,66,49,32)(16,84,67,50,33)(17,85,68,51,34), (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), (2,14,17,5)(3,10,16,9)(4,6,15,13)(7,11,12,8)(19,31,34,22)(20,27,33,26)(21,23,32,30)(24,28,29,25)(36,48,51,39)(37,44,50,43)(38,40,49,47)(41,45,46,42)(53,65,68,56)(54,61,67,60)(55,57,66,64)(58,62,63,59)(70,82,85,73)(71,78,84,77)(72,74,83,81)(75,79,80,76) );
G=PermutationGroup([[(1,69,52,35,18),(2,70,53,36,19),(3,71,54,37,20),(4,72,55,38,21),(5,73,56,39,22),(6,74,57,40,23),(7,75,58,41,24),(8,76,59,42,25),(9,77,60,43,26),(10,78,61,44,27),(11,79,62,45,28),(12,80,63,46,29),(13,81,64,47,30),(14,82,65,48,31),(15,83,66,49,32),(16,84,67,50,33),(17,85,68,51,34)], [(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)], [(2,14,17,5),(3,10,16,9),(4,6,15,13),(7,11,12,8),(19,31,34,22),(20,27,33,26),(21,23,32,30),(24,28,29,25),(36,48,51,39),(37,44,50,43),(38,40,49,47),(41,45,46,42),(53,65,68,56),(54,61,67,60),(55,57,66,64),(58,62,63,59),(70,82,85,73),(71,78,84,77),(72,74,83,81),(75,79,80,76)]])
40 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 17A | 17B | 17C | 17D | 20A | ··· | 20H | 85A | ··· | 85P |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 17 | 17 | 17 | 17 | 20 | ··· | 20 | 85 | ··· | 85 |
| size | 1 | 17 | 17 | 17 | 1 | 1 | 1 | 1 | 17 | 17 | 17 | 17 | 4 | 4 | 4 | 4 | 17 | ··· | 17 | 4 | ··· | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | |||||
| image | C1 | C2 | C4 | C5 | C10 | C20 | C17⋊C4 | C5×C17⋊C4 |
| kernel | C5×C17⋊C4 | C5×D17 | C85 | C17⋊C4 | D17 | C17 | C5 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 4 | 16 |
Matrix representation of C5×C17⋊C4 ►in GL4(𝔽1021) generated by
| 589 | 0 | 0 | 0 |
| 0 | 589 | 0 | 0 |
| 0 | 0 | 589 | 0 |
| 0 | 0 | 0 | 589 |
| 164 | 895 | 40 | 251 |
| 1 | 0 | 0 | 77 |
| 0 | 1 | 0 | 23 |
| 0 | 0 | 1 | 733 |
| 828 | 5 | 742 | 795 |
| 584 | 765 | 587 | 774 |
| 263 | 501 | 239 | 996 |
| 378 | 996 | 462 | 210 |
G:=sub<GL(4,GF(1021))| [589,0,0,0,0,589,0,0,0,0,589,0,0,0,0,589],[164,1,0,0,895,0,1,0,40,0,0,1,251,77,23,733],[828,584,263,378,5,765,501,996,742,587,239,462,795,774,996,210] >;
C5×C17⋊C4 in GAP, Magma, Sage, TeX
C_5\times C_{17}\rtimes C_4 % in TeX
G:=Group("C5xC17:C4"); // GroupNames label
G:=SmallGroup(340,5);
// by ID
G=gap.SmallGroup(340,5);
# by ID
G:=PCGroup([4,-2,-5,-2,-17,40,4163,523]);
// Polycyclic
G:=Group<a,b,c|a^5=b^17=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
Export