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G = C4xC20order 80 = 24·5

Abelian group of type [4,20]

direct product, abelian, monomial, 2-elementary

Aliases: C4xC20, SmallGroup(80,20)

Series: Derived Chief Lower central Upper central

C1 — C4xC20
C1C2C22C2xC10C2xC20 — C4xC20
C1 — C4xC20
C1 — C4xC20

Generators and relations for C4xC20
 G = < a,b | a4=b20=1, ab=ba >

Subgroups: 30, all normal (6 characteristic)
Quotients: C1, C2, C4, C22, C5, C2xC4, C10, C42, C20, C2xC10, C2xC20, C4xC20

Smallest permutation representation of C4xC20
Regular action on 80 points
Generators in S80
(1 80 46 32)(2 61 47 33)(3 62 48 34)(4 63 49 35)(5 64 50 36)(6 65 51 37)(7 66 52 38)(8 67 53 39)(9 68 54 40)(10 69 55 21)(11 70 56 22)(12 71 57 23)(13 72 58 24)(14 73 59 25)(15 74 60 26)(16 75 41 27)(17 76 42 28)(18 77 43 29)(19 78 44 30)(20 79 45 31)
(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)| (1,80,46,32)(2,61,47,33)(3,62,48,34)(4,63,49,35)(5,64,50,36)(6,65,51,37)(7,66,52,38)(8,67,53,39)(9,68,54,40)(10,69,55,21)(11,70,56,22)(12,71,57,23)(13,72,58,24)(14,73,59,25)(15,74,60,26)(16,75,41,27)(17,76,42,28)(18,77,43,29)(19,78,44,30)(20,79,45,31), (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( (1,80,46,32)(2,61,47,33)(3,62,48,34)(4,63,49,35)(5,64,50,36)(6,65,51,37)(7,66,52,38)(8,67,53,39)(9,68,54,40)(10,69,55,21)(11,70,56,22)(12,71,57,23)(13,72,58,24)(14,73,59,25)(15,74,60,26)(16,75,41,27)(17,76,42,28)(18,77,43,29)(19,78,44,30)(20,79,45,31), (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([[(1,80,46,32),(2,61,47,33),(3,62,48,34),(4,63,49,35),(5,64,50,36),(6,65,51,37),(7,66,52,38),(8,67,53,39),(9,68,54,40),(10,69,55,21),(11,70,56,22),(12,71,57,23),(13,72,58,24),(14,73,59,25),(15,74,60,26),(16,75,41,27),(17,76,42,28),(18,77,43,29),(19,78,44,30),(20,79,45,31)], [(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)]])

C4xC20 is a maximal subgroup of
C42.D5  C20:3C8  D20:4C4  C20:2Q8  C20.6Q8  C42:D5  C20:4D4  C4.D20  C42:2D5

80 conjugacy classes

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

80 irreducible representations

dim111111
type++
imageC1C2C4C5C10C20
kernelC4xC20C2xC20C20C42C2xC4C4
# reps131241248

Matrix representation of C4xC20 in GL2(F41) generated by

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

C4xC20 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 C4xC20 in TeX

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