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G = C4×C20order 80 = 24·5

Abelian group of type [4,20]

direct product, abelian, monomial, 2-elementary

Aliases: C4×C20, SmallGroup(80,20)

Series: Derived Chief Lower central Upper central

C1 — C4×C20
C1C2C22C2×C10C2×C20 — C4×C20
C1 — C4×C20
C1 — C4×C20

Generators and relations for C4×C20
 G = < a,b | a4=b20=1, ab=ba >


Smallest permutation representation of C4×C20
Regular action on 80 points
Generators in S80
(1 39 65 56)(2 40 66 57)(3 21 67 58)(4 22 68 59)(5 23 69 60)(6 24 70 41)(7 25 71 42)(8 26 72 43)(9 27 73 44)(10 28 74 45)(11 29 75 46)(12 30 76 47)(13 31 77 48)(14 32 78 49)(15 33 79 50)(16 34 80 51)(17 35 61 52)(18 36 62 53)(19 37 63 54)(20 38 64 55)
(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)

G:=sub<Sym(80)| (1,39,65,56)(2,40,66,57)(3,21,67,58)(4,22,68,59)(5,23,69,60)(6,24,70,41)(7,25,71,42)(8,26,72,43)(9,27,73,44)(10,28,74,45)(11,29,75,46)(12,30,76,47)(13,31,77,48)(14,32,78,49)(15,33,79,50)(16,34,80,51)(17,35,61,52)(18,36,62,53)(19,37,63,54)(20,38,64,55), (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)>;

G:=Group( (1,39,65,56)(2,40,66,57)(3,21,67,58)(4,22,68,59)(5,23,69,60)(6,24,70,41)(7,25,71,42)(8,26,72,43)(9,27,73,44)(10,28,74,45)(11,29,75,46)(12,30,76,47)(13,31,77,48)(14,32,78,49)(15,33,79,50)(16,34,80,51)(17,35,61,52)(18,36,62,53)(19,37,63,54)(20,38,64,55), (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) );

G=PermutationGroup([(1,39,65,56),(2,40,66,57),(3,21,67,58),(4,22,68,59),(5,23,69,60),(6,24,70,41),(7,25,71,42),(8,26,72,43),(9,27,73,44),(10,28,74,45),(11,29,75,46),(12,30,76,47),(13,31,77,48),(14,32,78,49),(15,33,79,50),(16,34,80,51),(17,35,61,52),(18,36,62,53),(19,37,63,54),(20,38,64,55)], [(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)])

C4×C20 is a maximal subgroup of
C42.D5  C203C8  D204C4  C202Q8  C20.6Q8  C42⋊D5  C204D4  C4.D20  C422D5

80 conjugacy classes

class 1 2A2B2C4A···4L5A5B5C5D10A···10L20A···20AV
order12224···4555510···1020···20
size11111···111111···11···1

80 irreducible representations

dim111111
type++
imageC1C2C4C5C10C20
kernelC4×C20C2×C20C20C42C2×C4C4
# reps131241248

Matrix representation of C4×C20 in GL2(𝔽41) generated by

400
032
,
80
016
G:=sub<GL(2,GF(41))| [40,0,0,32],[8,0,0,16] >;

C4×C20 in GAP, Magma, Sage, TeX

C_4\times C_{20}
% in TeX

G:=Group("C4xC20");
// GroupNames label

G:=SmallGroup(80,20);
// by ID

G=gap.SmallGroup(80,20);
# by ID

G:=PCGroup([5,-2,-2,-5,-2,-2,100,206]);
// Polycyclic

G:=Group<a,b|a^4=b^20=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C4×C20 in TeX

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