direct product, abelian, monomial, 5-elementary
Aliases: C5xC20, SmallGroup(100,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C5xC20 |
| C1 — C5xC20 |
| C1 — C5xC20 |
Generators and relations for C5xC20
G = < a,b | a5=b20=1, ab=ba >
Quotients: C1, C2, C4, C5, C10, C20, C52, C5xC10, C5xC20
Smallest permutation representation of C5xC20
►Regular action on 100 points
Generators in S100
(1 88 23 59 68)(2 89 24 60 69)(3 90 25 41 70)(4 91 26 42 71)(5 92 27 43 72)(6 93 28 44 73)(7 94 29 45 74)(8 95 30 46 75)(9 96 31 47 76)(10 97 32 48 77)(11 98 33 49 78)(12 99 34 50 79)(13 100 35 51 80)(14 81 36 52 61)(15 82 37 53 62)(16 83 38 54 63)(17 84 39 55 64)(18 85 40 56 65)(19 86 21 57 66)(20 87 22 58 67)
(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)
G:=sub<Sym(100)| (1,88,23,59,68)(2,89,24,60,69)(3,90,25,41,70)(4,91,26,42,71)(5,92,27,43,72)(6,93,28,44,73)(7,94,29,45,74)(8,95,30,46,75)(9,96,31,47,76)(10,97,32,48,77)(11,98,33,49,78)(12,99,34,50,79)(13,100,35,51,80)(14,81,36,52,61)(15,82,37,53,62)(16,83,38,54,63)(17,84,39,55,64)(18,85,40,56,65)(19,86,21,57,66)(20,87,22,58,67), (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)>;
G:=Group( (1,88,23,59,68)(2,89,24,60,69)(3,90,25,41,70)(4,91,26,42,71)(5,92,27,43,72)(6,93,28,44,73)(7,94,29,45,74)(8,95,30,46,75)(9,96,31,47,76)(10,97,32,48,77)(11,98,33,49,78)(12,99,34,50,79)(13,100,35,51,80)(14,81,36,52,61)(15,82,37,53,62)(16,83,38,54,63)(17,84,39,55,64)(18,85,40,56,65)(19,86,21,57,66)(20,87,22,58,67), (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) );
G=PermutationGroup([[(1,88,23,59,68),(2,89,24,60,69),(3,90,25,41,70),(4,91,26,42,71),(5,92,27,43,72),(6,93,28,44,73),(7,94,29,45,74),(8,95,30,46,75),(9,96,31,47,76),(10,97,32,48,77),(11,98,33,49,78),(12,99,34,50,79),(13,100,35,51,80),(14,81,36,52,61),(15,82,37,53,62),(16,83,38,54,63),(17,84,39,55,64),(18,85,40,56,65),(19,86,21,57,66),(20,87,22,58,67)], [(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)]])
C5xC20 is a maximal subgroup of
C52:7C8 C52:4Q8 C20:D5
100 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | ··· | 5X | 10A | ··· | 10X | 20A | ··· | 20AV |
| order | 1 | 2 | 4 | 4 | 5 | ··· | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||
| image | C1 | C2 | C4 | C5 | C10 | C20 |
| kernel | C5xC20 | C5xC10 | C52 | C20 | C10 | C5 |
| # reps | 1 | 1 | 2 | 24 | 24 | 48 |
Matrix representation of C5xC20 ►in GL2(F41) generated by
| 16 | 0 |
| 0 | 10 |
| 39 | 0 |
| 0 | 31 |
G:=sub<GL(2,GF(41))| [16,0,0,10],[39,0,0,31] >;
C5xC20 in GAP, Magma, Sage, TeX
C_5\times C_{20} % in TeX
G:=Group("C5xC20"); // GroupNames label
G:=SmallGroup(100,8);
// by ID
G=gap.SmallGroup(100,8);
# by ID
G:=PCGroup([4,-2,-5,-5,-2,200]);
// Polycyclic
G:=Group<a,b|a^5=b^20=1,a*b=b*a>;
// generators/relations
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