direct product, abelian, monomial, 5-elementary
Aliases: C5×C20, SmallGroup(100,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C5×C20 |
| C1 — C5×C20 |
| C1 — C5×C20 |
Generators and relations for C5×C20
G = < a,b | a5=b20=1, ab=ba >
Smallest permutation representation of C5×C20
►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)]])
C5×C20 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 | C5×C20 | C5×C10 | C52 | C20 | C10 | C5 |
| # reps | 1 | 1 | 2 | 24 | 24 | 48 |
Matrix representation of C5×C20 ►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] >;
C5×C20 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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