metacyclic, supersoluble, monomial
Aliases: C50.C10, C25⋊2C20, Dic25⋊C5, C52.Dic5, 5- 1+2⋊2C4, C2.(C25⋊C10), C10.3(C5×D5), (C5×C10).2D5, C5.3(C5×Dic5), (C2×5- 1+2).C2, SmallGroup(500,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C25 — C50.C10 |
Generators and relations for C50.C10
G = < a,b | a50=1, b10=a25, bab-1=a9 >
Smallest permutation representation of C50.C10
►On 100 points
Generators in S100
(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 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)
(1 62 26 87)(2 51 47 56 42 61 37 66 32 71 27 76 22 81 17 86 12 91 7 96)(3 90 18 75 33 60 48 95 13 80 28 65 43 100 8 85 23 70 38 55)(4 79 39 94 24 59 9 74 44 89 29 54 14 69 49 84 34 99 19 64)(5 68 10 63 15 58 20 53 25 98 30 93 35 88 40 83 45 78 50 73)(6 57 31 82)(11 52 36 77)(16 97 41 72)(21 92 46 67)
G:=sub<Sym(100)| (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)(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,62,26,87)(2,51,47,56,42,61,37,66,32,71,27,76,22,81,17,86,12,91,7,96)(3,90,18,75,33,60,48,95,13,80,28,65,43,100,8,85,23,70,38,55)(4,79,39,94,24,59,9,74,44,89,29,54,14,69,49,84,34,99,19,64)(5,68,10,63,15,58,20,53,25,98,30,93,35,88,40,83,45,78,50,73)(6,57,31,82)(11,52,36,77)(16,97,41,72)(21,92,46,67)>;
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)(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,62,26,87)(2,51,47,56,42,61,37,66,32,71,27,76,22,81,17,86,12,91,7,96)(3,90,18,75,33,60,48,95,13,80,28,65,43,100,8,85,23,70,38,55)(4,79,39,94,24,59,9,74,44,89,29,54,14,69,49,84,34,99,19,64)(5,68,10,63,15,58,20,53,25,98,30,93,35,88,40,83,45,78,50,73)(6,57,31,82)(11,52,36,77)(16,97,41,72)(21,92,46,67) );
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),(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)], [(1,62,26,87),(2,51,47,56,42,61,37,66,32,71,27,76,22,81,17,86,12,91,7,96),(3,90,18,75,33,60,48,95,13,80,28,65,43,100,8,85,23,70,38,55),(4,79,39,94,24,59,9,74,44,89,29,54,14,69,49,84,34,99,19,64),(5,68,10,63,15,58,20,53,25,98,30,93,35,88,40,83,45,78,50,73),(6,57,31,82),(11,52,36,77),(16,97,41,72),(21,92,46,67)]])
44 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 5E | 5F | 10A | 10B | 10C | 10D | 10E | 10F | 20A | ··· | 20H | 25A | ··· | 25J | 50A | ··· | 50J |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 25 | ··· | 25 | 50 | ··· | 50 |
| size | 1 | 1 | 25 | 25 | 2 | 2 | 5 | 5 | 5 | 5 | 2 | 2 | 5 | 5 | 5 | 5 | 25 | ··· | 25 | 10 | ··· | 10 | 10 | ··· | 10 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | ||||||
| image | C1 | C2 | C4 | C5 | C10 | C20 | C25⋊C10 | C50.C10 | D5 | Dic5 | C5×D5 | C5×Dic5 |
| kernel | C50.C10 | C2×5- 1+2 | 5- 1+2 | Dic25 | C50 | C25 | C2 | C1 | C5×C10 | C52 | C10 | C5 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 8 | 8 |
Matrix representation of C50.C10 ►in GL10(𝔽101)
| 79 | 79 | 21 | 21 | 80 | 80 | 23 | 44 | 0 | 0 |
| 1 | 1 | 22 | 22 | 79 | 79 | 78 | 21 | 0 | 0 |
| 80 | 80 | 21 | 21 | 79 | 79 | 0 | 0 | 23 | 44 |
| 79 | 79 | 22 | 22 | 1 | 1 | 0 | 0 | 78 | 21 |
| 79 | 100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 22 | 0 | 1 | 0 | 0 | 0 | 22 | 0 | 80 |
| 0 | 21 | 79 | 78 | 0 | 0 | 0 | 21 | 0 | 58 |
| 0 | 100 | 0 | 0 | 0 | 1 | 0 | 21 | 0 | 80 |
| 0 | 22 | 0 | 0 | 79 | 78 | 0 | 43 | 0 | 58 |
| 26 | 80 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 13 | 75 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 30 | 30 | 71 | 71 | 17 | 17 | 0 | 0 | 9 | 63 |
| 43 | 43 | 58 | 58 | 21 | 21 | 0 | 0 | 93 | 54 |
| 88 | 88 | 17 | 17 | 84 | 84 | 30 | 4 | 0 | 0 |
| 9 | 9 | 97 | 97 | 4 | 4 | 12 | 75 | 0 | 0 |
| 80 | 54 | 34 | 34 | 62 | 88 | 26 | 97 | 9 | 43 |
| 58 | 24 | 60 | 60 | 76 | 63 | 34 | 88 | 4 | 64 |
| 84 | 50 | 26 | 43 | 75 | 75 | 8 | 92 | 9 | 43 |
| 71 | 11 | 51 | 55 | 67 | 67 | 26 | 97 | 4 | 64 |
G:=sub<GL(10,GF(101))| [79,1,80,79,79,1,0,0,0,0,79,1,80,79,100,0,22,21,100,22,21,22,21,22,0,0,0,79,0,0,21,22,21,22,0,0,1,78,0,0,80,79,79,1,0,0,0,0,0,79,80,79,79,1,0,0,0,0,1,78,23,78,0,0,0,0,0,0,0,0,44,21,0,0,0,0,22,21,21,43,0,0,23,78,0,0,0,0,0,0,0,0,44,21,0,0,80,58,80,58],[26,13,30,43,88,9,80,58,84,71,80,75,30,43,88,9,54,24,50,11,0,0,71,58,17,97,34,60,26,51,0,0,71,58,17,97,34,60,43,55,0,0,17,21,84,4,62,76,75,67,0,0,17,21,84,4,88,63,75,67,0,0,0,0,30,12,26,34,8,26,0,0,0,0,4,75,97,88,92,97,0,0,9,93,0,0,9,4,9,4,0,0,63,54,0,0,43,64,43,64] >;
C50.C10 in GAP, Magma, Sage, TeX
C_{50}.C_{10} % in TeX
G:=Group("C50.C10"); // GroupNames label
G:=SmallGroup(500,9);
// by ID
G=gap.SmallGroup(500,9);
# by ID
G:=PCGroup([5,-2,-5,-2,-5,-5,50,3603,1208,418,10004]);
// Polycyclic
G:=Group<a,b|a^50=1,b^10=a^25,b*a*b^-1=a^9>;
// generators/relations
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