direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C5×Dic25, C25⋊5C20, 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
| C25 — C5×Dic25 |
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 68 26 93)(2 67 27 92)(3 66 28 91)(4 65 29 90)(5 64 30 89)(6 63 31 88)(7 62 32 87)(8 61 33 86)(9 60 34 85)(10 59 35 84)(11 58 36 83)(12 57 37 82)(13 56 38 81)(14 55 39 80)(15 54 40 79)(16 53 41 78)(17 52 42 77)(18 51 43 76)(19 100 44 75)(20 99 45 74)(21 98 46 73)(22 97 47 72)(23 96 48 71)(24 95 49 70)(25 94 50 69)
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), (1,68,26,93)(2,67,27,92)(3,66,28,91)(4,65,29,90)(5,64,30,89)(6,63,31,88)(7,62,32,87)(8,61,33,86)(9,60,34,85)(10,59,35,84)(11,58,36,83)(12,57,37,82)(13,56,38,81)(14,55,39,80)(15,54,40,79)(16,53,41,78)(17,52,42,77)(18,51,43,76)(19,100,44,75)(20,99,45,74)(21,98,46,73)(22,97,47,72)(23,96,48,71)(24,95,49,70)(25,94,50,69)>;
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), (1,68,26,93)(2,67,27,92)(3,66,28,91)(4,65,29,90)(5,64,30,89)(6,63,31,88)(7,62,32,87)(8,61,33,86)(9,60,34,85)(10,59,35,84)(11,58,36,83)(12,57,37,82)(13,56,38,81)(14,55,39,80)(15,54,40,79)(16,53,41,78)(17,52,42,77)(18,51,43,76)(19,100,44,75)(20,99,45,74)(21,98,46,73)(22,97,47,72)(23,96,48,71)(24,95,49,70)(25,94,50,69) );
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)], [(1,68,26,93),(2,67,27,92),(3,66,28,91),(4,65,29,90),(5,64,30,89),(6,63,31,88),(7,62,32,87),(8,61,33,86),(9,60,34,85),(10,59,35,84),(11,58,36,83),(12,57,37,82),(13,56,38,81),(14,55,39,80),(15,54,40,79),(16,53,41,78),(17,52,42,77),(18,51,43,76),(19,100,44,75),(20,99,45,74),(21,98,46,73),(22,97,47,72),(23,96,48,71),(24,95,49,70),(25,94,50,69)]])
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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