direct product, metacyclic, supersoluble, monomial, A-group
Aliases: Dic5×C25, C10.C50, C5⋊2C100, C50.4D5, C52.3C20, (C5×C25)⋊7C4, C2.(D5×C25), (C5×C50).1C2, C10.8(C5×D5), (C5×Dic5).C5, (C5×C10).5C10, C5.4(C5×Dic5), SmallGroup(500,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — Dic5×C25 |
Generators and relations for Dic5×C25
G = < a,b,c | a25=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Dic5×C25
►On 100 points
Generators in S100
(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 53 16 68 6 58 21 73 11 63)(2 54 17 69 7 59 22 74 12 64)(3 55 18 70 8 60 23 75 13 65)(4 56 19 71 9 61 24 51 14 66)(5 57 20 72 10 62 25 52 15 67)(26 100 36 85 46 95 31 80 41 90)(27 76 37 86 47 96 32 81 42 91)(28 77 38 87 48 97 33 82 43 92)(29 78 39 88 49 98 34 83 44 93)(30 79 40 89 50 99 35 84 45 94)
(1 85 58 41)(2 86 59 42)(3 87 60 43)(4 88 61 44)(5 89 62 45)(6 90 63 46)(7 91 64 47)(8 92 65 48)(9 93 66 49)(10 94 67 50)(11 95 68 26)(12 96 69 27)(13 97 70 28)(14 98 71 29)(15 99 72 30)(16 100 73 31)(17 76 74 32)(18 77 75 33)(19 78 51 34)(20 79 52 35)(21 80 53 36)(22 81 54 37)(23 82 55 38)(24 83 56 39)(25 84 57 40)
G:=sub<Sym(100)| (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), (1,53,16,68,6,58,21,73,11,63)(2,54,17,69,7,59,22,74,12,64)(3,55,18,70,8,60,23,75,13,65)(4,56,19,71,9,61,24,51,14,66)(5,57,20,72,10,62,25,52,15,67)(26,100,36,85,46,95,31,80,41,90)(27,76,37,86,47,96,32,81,42,91)(28,77,38,87,48,97,33,82,43,92)(29,78,39,88,49,98,34,83,44,93)(30,79,40,89,50,99,35,84,45,94), (1,85,58,41)(2,86,59,42)(3,87,60,43)(4,88,61,44)(5,89,62,45)(6,90,63,46)(7,91,64,47)(8,92,65,48)(9,93,66,49)(10,94,67,50)(11,95,68,26)(12,96,69,27)(13,97,70,28)(14,98,71,29)(15,99,72,30)(16,100,73,31)(17,76,74,32)(18,77,75,33)(19,78,51,34)(20,79,52,35)(21,80,53,36)(22,81,54,37)(23,82,55,38)(24,83,56,39)(25,84,57,40)>;
G:=Group( (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), (1,53,16,68,6,58,21,73,11,63)(2,54,17,69,7,59,22,74,12,64)(3,55,18,70,8,60,23,75,13,65)(4,56,19,71,9,61,24,51,14,66)(5,57,20,72,10,62,25,52,15,67)(26,100,36,85,46,95,31,80,41,90)(27,76,37,86,47,96,32,81,42,91)(28,77,38,87,48,97,33,82,43,92)(29,78,39,88,49,98,34,83,44,93)(30,79,40,89,50,99,35,84,45,94), (1,85,58,41)(2,86,59,42)(3,87,60,43)(4,88,61,44)(5,89,62,45)(6,90,63,46)(7,91,64,47)(8,92,65,48)(9,93,66,49)(10,94,67,50)(11,95,68,26)(12,96,69,27)(13,97,70,28)(14,98,71,29)(15,99,72,30)(16,100,73,31)(17,76,74,32)(18,77,75,33)(19,78,51,34)(20,79,52,35)(21,80,53,36)(22,81,54,37)(23,82,55,38)(24,83,56,39)(25,84,57,40) );
G=PermutationGroup([[(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)], [(1,53,16,68,6,58,21,73,11,63),(2,54,17,69,7,59,22,74,12,64),(3,55,18,70,8,60,23,75,13,65),(4,56,19,71,9,61,24,51,14,66),(5,57,20,72,10,62,25,52,15,67),(26,100,36,85,46,95,31,80,41,90),(27,76,37,86,47,96,32,81,42,91),(28,77,38,87,48,97,33,82,43,92),(29,78,39,88,49,98,34,83,44,93),(30,79,40,89,50,99,35,84,45,94)], [(1,85,58,41),(2,86,59,42),(3,87,60,43),(4,88,61,44),(5,89,62,45),(6,90,63,46),(7,91,64,47),(8,92,65,48),(9,93,66,49),(10,94,67,50),(11,95,68,26),(12,96,69,27),(13,97,70,28),(14,98,71,29),(15,99,72,30),(16,100,73,31),(17,76,74,32),(18,77,75,33),(19,78,51,34),(20,79,52,35),(21,80,53,36),(22,81,54,37),(23,82,55,38),(24,83,56,39),(25,84,57,40)]])
200 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 10A | 10B | 10C | 10D | 10E | ··· | 10N | 20A | ··· | 20H | 25A | ··· | 25T | 25U | ··· | 25BH | 50A | ··· | 50T | 50U | ··· | 50BH | 100A | ··· | 100AN |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 25 | ··· | 25 | 25 | ··· | 25 | 50 | ··· | 50 | 50 | ··· | 50 | 100 | ··· | 100 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 5 | ··· | 5 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 |
200 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | |||||||||||
| image | C1 | C2 | C4 | C5 | C10 | C20 | C25 | C50 | C100 | D5 | Dic5 | C5×D5 | C5×Dic5 | D5×C25 | Dic5×C25 |
| kernel | Dic5×C25 | C5×C50 | C5×C25 | C5×Dic5 | C5×C10 | C52 | Dic5 | C10 | C5 | C50 | C25 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 20 | 20 | 40 | 2 | 2 | 8 | 8 | 40 | 40 |
Matrix representation of Dic5×C25 ►in GL2(𝔽101) generated by
| 54 | 0 |
| 0 | 54 |
| 14 | 0 |
| 0 | 65 |
| 0 | 1 |
| 100 | 0 |
G:=sub<GL(2,GF(101))| [54,0,0,54],[14,0,0,65],[0,100,1,0] >;
Dic5×C25 in GAP, Magma, Sage, TeX
{\rm Dic}_5\times C_{25} % in TeX
G:=Group("Dic5xC25"); // GroupNames label
G:=SmallGroup(500,7);
// by ID
G=gap.SmallGroup(500,7);
# by ID
G:=PCGroup([5,-2,-5,-2,-5,-5,50,106,10004]);
// Polycyclic
G:=Group<a,b,c|a^25=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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