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G = C5×Dic20order 400 = 24·52

Direct product of C5 and Dic20

direct product, metacyclic, supersoluble, monomial

Aliases: C5×Dic20, C40.9D5, C40.1C10, C524Q16, C10.21D20, C20.62D10, Dic10.1C10, C8.(C5×D5), C51(C5×Q16), (C5×C40).2C2, C10.3(C5×D4), C2.5(C5×D20), C4.10(D5×C10), (C5×C10).19D4, C20.10(C2×C10), (C5×C20).39C22, (C5×Dic10).4C2, SmallGroup(400,80)

Series: Derived Chief Lower central Upper central

C1C20 — C5×Dic20
C1C5C10C20C5×C20C5×Dic10 — C5×Dic20
C5C10C20 — C5×Dic20
C1C10C20C40

Generators and relations for C5×Dic20
 G = < a,b,c | a5=b40=1, c2=b20, ab=ba, ac=ca, cbc-1=b-1 >

2C5
2C5
10C4
10C4
2C10
2C10
5Q8
5Q8
2C20
2Dic5
2C20
2Dic5
10C20
10C20
5Q16
2C40
2C40
5C5×Q8
5C5×Q8
2C5×Dic5
2C5×Dic5
5C5×Q16

Smallest permutation representation of C5×Dic20
On 80 points
Generators in S80
(1 25 9 33 17)(2 26 10 34 18)(3 27 11 35 19)(4 28 12 36 20)(5 29 13 37 21)(6 30 14 38 22)(7 31 15 39 23)(8 32 16 40 24)(41 57 73 49 65)(42 58 74 50 66)(43 59 75 51 67)(44 60 76 52 68)(45 61 77 53 69)(46 62 78 54 70)(47 63 79 55 71)(48 64 80 56 72)
(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)
(1 41 21 61)(2 80 22 60)(3 79 23 59)(4 78 24 58)(5 77 25 57)(6 76 26 56)(7 75 27 55)(8 74 28 54)(9 73 29 53)(10 72 30 52)(11 71 31 51)(12 70 32 50)(13 69 33 49)(14 68 34 48)(15 67 35 47)(16 66 36 46)(17 65 37 45)(18 64 38 44)(19 63 39 43)(20 62 40 42)

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

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

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

115 conjugacy classes

class 1  2 4A4B4C5A5B5C5D5E···5N8A8B10A10B10C10D10E···10N20A···20X20Y···20AF40A···40AV
order1244455555···5881010101010···1020···2020···2040···40
size112202011112···22211112···22···220···202···2

115 irreducible representations

dim111111222222222222
type+++++-++-
imageC1C2C2C5C10C10D4D5Q16D10D20C5×D4C5×D5Dic20C5×Q16D5×C10C5×D20C5×Dic20
kernelC5×Dic20C5×C40C5×Dic10Dic20C40Dic10C5×C10C40C52C20C10C10C8C5C5C4C2C1
# reps11244812224488881632

Matrix representation of C5×Dic20 in GL2(𝔽41) generated by

180
018
,
340
035
,
01
400
G:=sub<GL(2,GF(41))| [18,0,0,18],[34,0,0,35],[0,40,1,0] >;

C5×Dic20 in GAP, Magma, Sage, TeX

C_5\times {\rm Dic}_{20}
% in TeX

G:=Group("C5xDic20");
// GroupNames label

G:=SmallGroup(400,80);
// by ID

G=gap.SmallGroup(400,80);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,240,265,367,1443,69,11525]);
// Polycyclic

G:=Group<a,b,c|a^5=b^40=1,c^2=b^20,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C5×Dic20 in TeX

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