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## G = C10×D25order 500 = 22·53

### Direct product of C10 and D25

Aliases: C10×D25, C503C10, C52.3D10, (C5×C50)⋊2C2, C253(C2×C10), (C5×C25)⋊3C22, (C5×C10).7D5, C5.1(D5×C10), C10.4(C5×D5), SmallGroup(500,28)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C25 — C10×D25
 Chief series C1 — C5 — C25 — C5×C25 — C5×D25 — C10×D25
 Lower central C25 — C10×D25
 Upper central C1 — C10

Generators and relations for C10×D25
G = < a,b,c | a10=b25=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C10×D25
On 100 points
Generators in S100
(1 32 11 42 21 27 6 37 16 47)(2 33 12 43 22 28 7 38 17 48)(3 34 13 44 23 29 8 39 18 49)(4 35 14 45 24 30 9 40 19 50)(5 36 15 46 25 31 10 41 20 26)(51 87 66 77 56 92 71 82 61 97)(52 88 67 78 57 93 72 83 62 98)(53 89 68 79 58 94 73 84 63 99)(54 90 69 80 59 95 74 85 64 100)(55 91 70 81 60 96 75 86 65 76)
(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 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)
(1 59)(2 58)(3 57)(4 56)(5 55)(6 54)(7 53)(8 52)(9 51)(10 75)(11 74)(12 73)(13 72)(14 71)(15 70)(16 69)(17 68)(18 67)(19 66)(20 65)(21 64)(22 63)(23 62)(24 61)(25 60)(26 76)(27 100)(28 99)(29 98)(30 97)(31 96)(32 95)(33 94)(34 93)(35 92)(36 91)(37 90)(38 89)(39 88)(40 87)(41 86)(42 85)(43 84)(44 83)(45 82)(46 81)(47 80)(48 79)(49 78)(50 77)

G:=sub<Sym(100)| (1,32,11,42,21,27,6,37,16,47)(2,33,12,43,22,28,7,38,17,48)(3,34,13,44,23,29,8,39,18,49)(4,35,14,45,24,30,9,40,19,50)(5,36,15,46,25,31,10,41,20,26)(51,87,66,77,56,92,71,82,61,97)(52,88,67,78,57,93,72,83,62,98)(53,89,68,79,58,94,73,84,63,99)(54,90,69,80,59,95,74,85,64,100)(55,91,70,81,60,96,75,86,65,76), (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)(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,59)(2,58)(3,57)(4,56)(5,55)(6,54)(7,53)(8,52)(9,51)(10,75)(11,74)(12,73)(13,72)(14,71)(15,70)(16,69)(17,68)(18,67)(19,66)(20,65)(21,64)(22,63)(23,62)(24,61)(25,60)(26,76)(27,100)(28,99)(29,98)(30,97)(31,96)(32,95)(33,94)(34,93)(35,92)(36,91)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,78)(50,77)>;

G:=Group( (1,32,11,42,21,27,6,37,16,47)(2,33,12,43,22,28,7,38,17,48)(3,34,13,44,23,29,8,39,18,49)(4,35,14,45,24,30,9,40,19,50)(5,36,15,46,25,31,10,41,20,26)(51,87,66,77,56,92,71,82,61,97)(52,88,67,78,57,93,72,83,62,98)(53,89,68,79,58,94,73,84,63,99)(54,90,69,80,59,95,74,85,64,100)(55,91,70,81,60,96,75,86,65,76), (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)(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,59)(2,58)(3,57)(4,56)(5,55)(6,54)(7,53)(8,52)(9,51)(10,75)(11,74)(12,73)(13,72)(14,71)(15,70)(16,69)(17,68)(18,67)(19,66)(20,65)(21,64)(22,63)(23,62)(24,61)(25,60)(26,76)(27,100)(28,99)(29,98)(30,97)(31,96)(32,95)(33,94)(34,93)(35,92)(36,91)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,78)(50,77) );

G=PermutationGroup([(1,32,11,42,21,27,6,37,16,47),(2,33,12,43,22,28,7,38,17,48),(3,34,13,44,23,29,8,39,18,49),(4,35,14,45,24,30,9,40,19,50),(5,36,15,46,25,31,10,41,20,26),(51,87,66,77,56,92,71,82,61,97),(52,88,67,78,57,93,72,83,62,98),(53,89,68,79,58,94,73,84,63,99),(54,90,69,80,59,95,74,85,64,100),(55,91,70,81,60,96,75,86,65,76)], [(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),(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)], [(1,59),(2,58),(3,57),(4,56),(5,55),(6,54),(7,53),(8,52),(9,51),(10,75),(11,74),(12,73),(13,72),(14,71),(15,70),(16,69),(17,68),(18,67),(19,66),(20,65),(21,64),(22,63),(23,62),(24,61),(25,60),(26,76),(27,100),(28,99),(29,98),(30,97),(31,96),(32,95),(33,94),(34,93),(35,92),(36,91),(37,90),(38,89),(39,88),(40,87),(41,86),(42,85),(43,84),(44,83),(45,82),(46,81),(47,80),(48,79),(49,78),(50,77)])

140 conjugacy classes

 class 1 2A 2B 2C 5A 5B 5C 5D 5E ··· 5N 10A 10B 10C 10D 10E ··· 10N 10O ··· 10V 25A ··· 25AX 50A ··· 50AX order 1 2 2 2 5 5 5 5 5 ··· 5 10 10 10 10 10 ··· 10 10 ··· 10 25 ··· 25 50 ··· 50 size 1 1 25 25 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 25 ··· 25 2 ··· 2 2 ··· 2

140 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C5 C10 C10 D5 D10 D25 C5×D5 D50 D5×C10 C5×D25 C10×D25 kernel C10×D25 C5×D25 C5×C50 D50 D25 C50 C5×C10 C52 C10 C10 C5 C5 C2 C1 # reps 1 2 1 4 8 4 2 2 10 8 10 8 40 40

Matrix representation of C10×D25 in GL3(𝔽101) generated by

 65 0 0 0 84 0 0 0 84
,
 1 0 0 0 88 20 0 0 31
,
 1 0 0 0 88 20 0 32 13
G:=sub<GL(3,GF(101))| [65,0,0,0,84,0,0,0,84],[1,0,0,0,88,0,0,20,31],[1,0,0,0,88,32,0,20,13] >;

C10×D25 in GAP, Magma, Sage, TeX

C_{10}\times D_{25}
% in TeX

G:=Group("C10xD25");
// GroupNames label

G:=SmallGroup(500,28);
// by ID

G=gap.SmallGroup(500,28);
# by ID

G:=PCGroup([5,-2,-2,-5,-5,-5,3603,418,10004]);
// Polycyclic

G:=Group<a,b,c|a^10=b^25=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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