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G = C5×Dic25order 500 = 22·53

Direct product of C5 and Dic25

direct product, metacyclic, supersoluble, monomial, A-group

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

C1C25 — C5×Dic25
C1C5C25C50C5×C50 — C5×Dic25
C25 — C5×Dic25
C1C10

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

2C5
2C5
25C4
2C10
2C10
2C25
2C25
5Dic5
25C20
2C50
2C50
5C5×Dic5

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 4A4B5A5B5C5D5E···5N10A10B10C10D10E···10N20A···20H25A···25AX50A···50AX
order124455555···51010101010···1020···2025···2550···50
size11252511112···211112···225···252···22···2

140 irreducible representations

dim11111122222222
type+++-+-
imageC1C2C4C5C10C20D5Dic5D25C5×D5Dic25C5×Dic5C5×D25C5×Dic25
kernelC5×Dic25C5×C50C5×C25Dic25C50C25C5×C10C52C10C10C5C5C2C1
# reps112448221081084040

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

3600
0870
0087
,
10000
0810
005
,
9100
001
010
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

Export

Subgroup lattice of C5×Dic25 in TeX

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