direct product, metacyclic, supersoluble, monomial, A-group
Aliases: D5×C50, C10⋊C50, C5⋊(C2×C50), (C5×C50)⋊1C2, (D5×C10).C5, (C5×C25)⋊2C22, C10.9(C5×D5), C5.4(D5×C10), (C5×C10).6C10, (C5×D5).4C10, C52.2(C2×C10), SmallGroup(500,29)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C50 |
Generators and relations for D5×C50
G = < a,b,c | a50=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C50
►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 21 41 11 31)(2 22 42 12 32)(3 23 43 13 33)(4 24 44 14 34)(5 25 45 15 35)(6 26 46 16 36)(7 27 47 17 37)(8 28 48 18 38)(9 29 49 19 39)(10 30 50 20 40)(51 81 61 91 71)(52 82 62 92 72)(53 83 63 93 73)(54 84 64 94 74)(55 85 65 95 75)(56 86 66 96 76)(57 87 67 97 77)(58 88 68 98 78)(59 89 69 99 79)(60 90 70 100 80)
(1 80)(2 81)(3 82)(4 83)(5 84)(6 85)(7 86)(8 87)(9 88)(10 89)(11 90)(12 91)(13 92)(14 93)(15 94)(16 95)(17 96)(18 97)(19 98)(20 99)(21 100)(22 51)(23 52)(24 53)(25 54)(26 55)(27 56)(28 57)(29 58)(30 59)(31 60)(32 61)(33 62)(34 63)(35 64)(36 65)(37 66)(38 67)(39 68)(40 69)(41 70)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)
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), (1,21,41,11,31)(2,22,42,12,32)(3,23,43,13,33)(4,24,44,14,34)(5,25,45,15,35)(6,26,46,16,36)(7,27,47,17,37)(8,28,48,18,38)(9,29,49,19,39)(10,30,50,20,40)(51,81,61,91,71)(52,82,62,92,72)(53,83,63,93,73)(54,84,64,94,74)(55,85,65,95,75)(56,86,66,96,76)(57,87,67,97,77)(58,88,68,98,78)(59,89,69,99,79)(60,90,70,100,80), (1,80)(2,81)(3,82)(4,83)(5,84)(6,85)(7,86)(8,87)(9,88)(10,89)(11,90)(12,91)(13,92)(14,93)(15,94)(16,95)(17,96)(18,97)(19,98)(20,99)(21,100)(22,51)(23,52)(24,53)(25,54)(26,55)(27,56)(28,57)(29,58)(30,59)(31,60)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)>;
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), (1,21,41,11,31)(2,22,42,12,32)(3,23,43,13,33)(4,24,44,14,34)(5,25,45,15,35)(6,26,46,16,36)(7,27,47,17,37)(8,28,48,18,38)(9,29,49,19,39)(10,30,50,20,40)(51,81,61,91,71)(52,82,62,92,72)(53,83,63,93,73)(54,84,64,94,74)(55,85,65,95,75)(56,86,66,96,76)(57,87,67,97,77)(58,88,68,98,78)(59,89,69,99,79)(60,90,70,100,80), (1,80)(2,81)(3,82)(4,83)(5,84)(6,85)(7,86)(8,87)(9,88)(10,89)(11,90)(12,91)(13,92)(14,93)(15,94)(16,95)(17,96)(18,97)(19,98)(20,99)(21,100)(22,51)(23,52)(24,53)(25,54)(26,55)(27,56)(28,57)(29,58)(30,59)(31,60)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79) );
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)], [(1,21,41,11,31),(2,22,42,12,32),(3,23,43,13,33),(4,24,44,14,34),(5,25,45,15,35),(6,26,46,16,36),(7,27,47,17,37),(8,28,48,18,38),(9,29,49,19,39),(10,30,50,20,40),(51,81,61,91,71),(52,82,62,92,72),(53,83,63,93,73),(54,84,64,94,74),(55,85,65,95,75),(56,86,66,96,76),(57,87,67,97,77),(58,88,68,98,78),(59,89,69,99,79),(60,90,70,100,80)], [(1,80),(2,81),(3,82),(4,83),(5,84),(6,85),(7,86),(8,87),(9,88),(10,89),(11,90),(12,91),(13,92),(14,93),(15,94),(16,95),(17,96),(18,97),(19,98),(20,99),(21,100),(22,51),(23,52),(24,53),(25,54),(26,55),(27,56),(28,57),(29,58),(30,59),(31,60),(32,61),(33,62),(34,63),(35,64),(36,65),(37,66),(38,67),(39,68),(40,69),(41,70),(42,71),(43,72),(44,73),(45,74),(46,75),(47,76),(48,77),(49,78),(50,79)]])
200 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 10A | 10B | 10C | 10D | 10E | ··· | 10N | 10O | ··· | 10V | 25A | ··· | 25T | 25U | ··· | 25BH | 50A | ··· | 50T | 50U | ··· | 50BH | 50BI | ··· | 50CV |
| order | 1 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 25 | ··· | 25 | 25 | ··· | 25 | 50 | ··· | 50 | 50 | ··· | 50 | 50 | ··· | 50 |
| 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 | C2 | C5 | C10 | C10 | C25 | C50 | C50 | D5 | D10 | C5×D5 | D5×C10 | D5×C25 | D5×C50 |
| kernel | D5×C50 | D5×C25 | C5×C50 | D5×C10 | C5×D5 | C5×C10 | D10 | D5 | C10 | C50 | C25 | C10 | C5 | C2 | C1 |
| # reps | 1 | 2 | 1 | 4 | 8 | 4 | 20 | 40 | 20 | 2 | 2 | 8 | 8 | 40 | 40 |
Matrix representation of D5×C50 ►in GL2(𝔽101) generated by
| 45 | 0 |
| 0 | 45 |
| 36 | 0 |
| 0 | 87 |
| 0 | 14 |
| 65 | 0 |
G:=sub<GL(2,GF(101))| [45,0,0,45],[36,0,0,87],[0,65,14,0] >;
D5×C50 in GAP, Magma, Sage, TeX
D_5\times C_{50} % in TeX
G:=Group("D5xC50"); // GroupNames label
G:=SmallGroup(500,29);
// by ID
G=gap.SmallGroup(500,29);
# by ID
G:=PCGroup([5,-2,-2,-5,-5,-5,87,10004]);
// Polycyclic
G:=Group<a,b,c|a^50=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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