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## G = C5×Dic25order 500 = 22·53

### Direct product of C5 and Dic25

Aliases: C5×Dic25, C255C20, C50.3C10, C10.4D25, C52.3Dic5, (C5×C25)⋊8C4, C2.(C5×D25), (C5×C50).2C2, C10.1(C5×D5), (C5×C10).5D5, C5.1(C5×Dic5), SmallGroup(500,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C25 — C5×Dic25
 Chief series C1 — C5 — C25 — C50 — C5×C50 — C5×Dic25
 Lower central C25 — C5×Dic25
 Upper central C1 — C10

Generators and relations for C5×Dic25
G = < a,b,c | a5=b50=1, c2=b25, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C5×Dic25
On 100 points
Generators in S100
(1 11 21 31 41)(2 12 22 32 42)(3 13 23 33 43)(4 14 24 34 44)(5 15 25 35 45)(6 16 26 36 46)(7 17 27 37 47)(8 18 28 38 48)(9 19 29 39 49)(10 20 30 40 50)(51 91 81 71 61)(52 92 82 72 62)(53 93 83 73 63)(54 94 84 74 64)(55 95 85 75 65)(56 96 86 76 66)(57 97 87 77 67)(58 98 88 78 68)(59 99 89 79 69)(60 100 90 80 70)
(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 78 26 53)(2 77 27 52)(3 76 28 51)(4 75 29 100)(5 74 30 99)(6 73 31 98)(7 72 32 97)(8 71 33 96)(9 70 34 95)(10 69 35 94)(11 68 36 93)(12 67 37 92)(13 66 38 91)(14 65 39 90)(15 64 40 89)(16 63 41 88)(17 62 42 87)(18 61 43 86)(19 60 44 85)(20 59 45 84)(21 58 46 83)(22 57 47 82)(23 56 48 81)(24 55 49 80)(25 54 50 79)

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

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

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

140 conjugacy classes

 class 1 2 4A 4B 5A 5B 5C 5D 5E ··· 5N 10A 10B 10C 10D 10E ··· 10N 20A ··· 20H 25A ··· 25AX 50A ··· 50AX order 1 2 4 4 5 5 5 5 5 ··· 5 10 10 10 10 10 ··· 10 20 ··· 20 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 C4 C5 C10 C20 D5 Dic5 D25 C5×D5 Dic25 C5×Dic5 C5×D25 C5×Dic25 kernel C5×Dic25 C5×C50 C5×C25 Dic25 C50 C25 C5×C10 C52 C10 C10 C5 C5 C2 C1 # reps 1 1 2 4 4 8 2 2 10 8 10 8 40 40

Matrix representation of C5×Dic25 in GL3(𝔽101) generated by

 36 0 0 0 87 0 0 0 87
,
 100 0 0 0 81 0 0 0 5
,
 91 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(101))| [36,0,0,0,87,0,0,0,87],[100,0,0,0,81,0,0,0,5],[91,0,0,0,0,1,0,1,0] >;

C5×Dic25 in GAP, Magma, Sage, TeX

C_5\times {\rm Dic}_{25}
% in TeX

G:=Group("C5xDic25");
// GroupNames label

G:=SmallGroup(500,6);
// by ID

G=gap.SmallGroup(500,6);
# by ID

G:=PCGroup([5,-2,-5,-2,-5,-5,50,3603,418,10004]);
// Polycyclic

G:=Group<a,b,c|a^5=b^50=1,c^2=b^25,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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