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## G = C5×D40order 400 = 24·52

### Direct product of C5 and D40

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

 Derived series C1 — C20 — C5×D40
 Chief series C1 — C5 — C10 — C20 — C5×C20 — C5×D20 — C5×D40
 Lower central C5 — C10 — C20 — C5×D40
 Upper central C1 — C10 — C20 — C40

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

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

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

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

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

115 conjugacy classes

 class 1 2A 2B 2C 4 5A 5B 5C 5D 5E ··· 5N 8A 8B 10A 10B 10C 10D 10E ··· 10N 10O ··· 10V 20A ··· 20X 40A ··· 40AV order 1 2 2 2 4 5 5 5 5 5 ··· 5 8 8 10 10 10 10 10 ··· 10 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 20 20 2 1 1 1 1 2 ··· 2 2 2 1 1 1 1 2 ··· 2 20 ··· 20 2 ··· 2 2 ··· 2

115 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C5 C10 C10 D4 D5 D8 D10 D20 C5×D4 C5×D5 D40 C5×D8 D5×C10 C5×D20 C5×D40 kernel C5×D40 C5×C40 C5×D20 D40 C40 D20 C5×C10 C40 C52 C20 C10 C10 C8 C5 C5 C4 C2 C1 # reps 1 1 2 4 4 8 1 2 2 2 4 4 8 8 8 8 16 32

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

 16 0 0 16
,
 30 0 0 26
,
 0 26 30 0
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

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