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G = C5×D40order 400 = 24·52

Direct product of C5 and D40

direct product, metacyclic, supersoluble, monomial

Aliases: C5×D40, C405D5, C401C10, C524D8, D201C10, C10.20D20, C20.61D10, C51(C5×D8), C81(C5×D5), (C5×C40)⋊2C2, C2.4(C5×D20), C10.2(C5×D4), C4.9(D5×C10), (C5×D20)⋊10C2, C20.9(C2×C10), (C5×C10).18D4, (C5×C20).38C22, SmallGroup(400,79)

Series: Derived Chief Lower central Upper central

C1C20 — C5×D40
C1C5C10C20C5×C20C5×D20 — C5×D40
C5C10C20 — C5×D40
C1C10C20C40

Generators and relations for C5×D40
 G = < a,b,c | a5=b40=c2=1, ab=ba, ac=ca, cbc=b-1 >

20C2
20C2
2C5
2C5
10C22
10C22
2C10
2C10
4D5
4D5
20C10
20C10
5D4
5D4
2C20
2D10
2D10
2C20
10C2×C10
10C2×C10
4C5×D5
4C5×D5
5D8
2C40
2C40
5C5×D4
5C5×D4
2D5×C10
2D5×C10
5C5×D8

Smallest permutation representation of C5×D40
On 80 points
Generators in S80
(1 9 17 25 33)(2 10 18 26 34)(3 11 19 27 35)(4 12 20 28 36)(5 13 21 29 37)(6 14 22 30 38)(7 15 23 31 39)(8 16 24 32 40)(41 73 65 57 49)(42 74 66 58 50)(43 75 67 59 51)(44 76 68 60 52)(45 77 69 61 53)(46 78 70 62 54)(47 79 71 63 55)(48 80 72 64 56)
(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 78)(2 77)(3 76)(4 75)(5 74)(6 73)(7 72)(8 71)(9 70)(10 69)(11 68)(12 67)(13 66)(14 65)(15 64)(16 63)(17 62)(18 61)(19 60)(20 59)(21 58)(22 57)(23 56)(24 55)(25 54)(26 53)(27 52)(28 51)(29 50)(30 49)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 80)(40 79)

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

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

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

115 conjugacy classes

class 1 2A2B2C 4 5A5B5C5D5E···5N8A8B10A10B10C10D10E···10N10O···10V20A···20X40A···40AV
order1222455555···5881010101010···1010···1020···2040···40
size112020211112···22211112···220···202···22···2

115 irreducible representations

dim111111222222222222
type+++++++++
imageC1C2C2C5C10C10D4D5D8D10D20C5×D4C5×D5D40C5×D8D5×C10C5×D20C5×D40
kernelC5×D40C5×C40C5×D20D40C40D20C5×C10C40C52C20C10C10C8C5C5C4C2C1
# reps11244812224488881632

Matrix representation of C5×D40 in GL2(𝔽41) generated by

160
016
,
300
026
,
026
300
G:=sub<GL(2,GF(41))| [16,0,0,16],[30,0,0,26],[0,30,26,0] >;

C5×D40 in GAP, Magma, Sage, TeX

C_5\times D_{40}
% in TeX

G:=Group("C5xD40");
// GroupNames label

G:=SmallGroup(400,79);
// by ID

G=gap.SmallGroup(400,79);
# by ID

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

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

Export

Subgroup lattice of C5×D40 in TeX

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