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G = C5×C20order 100 = 22·52

Abelian group of type [5,20]

direct product, abelian, monomial, 5-elementary

Aliases: C5×C20, SmallGroup(100,8)

Series: Derived Chief Lower central Upper central

C1 — C5×C20
C1C2C10C5×C10 — 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 98 43 32 66)(2 99 44 33 67)(3 100 45 34 68)(4 81 46 35 69)(5 82 47 36 70)(6 83 48 37 71)(7 84 49 38 72)(8 85 50 39 73)(9 86 51 40 74)(10 87 52 21 75)(11 88 53 22 76)(12 89 54 23 77)(13 90 55 24 78)(14 91 56 25 79)(15 92 57 26 80)(16 93 58 27 61)(17 94 59 28 62)(18 95 60 29 63)(19 96 41 30 64)(20 97 42 31 65)
(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)| (1,98,43,32,66)(2,99,44,33,67)(3,100,45,34,68)(4,81,46,35,69)(5,82,47,36,70)(6,83,48,37,71)(7,84,49,38,72)(8,85,50,39,73)(9,86,51,40,74)(10,87,52,21,75)(11,88,53,22,76)(12,89,54,23,77)(13,90,55,24,78)(14,91,56,25,79)(15,92,57,26,80)(16,93,58,27,61)(17,94,59,28,62)(18,95,60,29,63)(19,96,41,30,64)(20,97,42,31,65), (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( (1,98,43,32,66)(2,99,44,33,67)(3,100,45,34,68)(4,81,46,35,69)(5,82,47,36,70)(6,83,48,37,71)(7,84,49,38,72)(8,85,50,39,73)(9,86,51,40,74)(10,87,52,21,75)(11,88,53,22,76)(12,89,54,23,77)(13,90,55,24,78)(14,91,56,25,79)(15,92,57,26,80)(16,93,58,27,61)(17,94,59,28,62)(18,95,60,29,63)(19,96,41,30,64)(20,97,42,31,65), (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([(1,98,43,32,66),(2,99,44,33,67),(3,100,45,34,68),(4,81,46,35,69),(5,82,47,36,70),(6,83,48,37,71),(7,84,49,38,72),(8,85,50,39,73),(9,86,51,40,74),(10,87,52,21,75),(11,88,53,22,76),(12,89,54,23,77),(13,90,55,24,78),(14,91,56,25,79),(15,92,57,26,80),(16,93,58,27,61),(17,94,59,28,62),(18,95,60,29,63),(19,96,41,30,64),(20,97,42,31,65)], [(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   C527C8  C524Q8  C20⋊D5

100 conjugacy classes

class 1  2 4A4B5A···5X10A···10X20A···20AV
order12445···510···1020···20
size11111···11···11···1

100 irreducible representations

dim111111
type++
imageC1C2C4C5C10C20
kernelC5×C20C5×C10C52C20C10C5
# reps112242448

Matrix representation of C5×C20 in GL2(𝔽41) generated by

160
010
,
390
031
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

Export

Subgroup lattice of C5×C20 in TeX

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