direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C10×D25, C50⋊3C10, C52.3D10, (C5×C50)⋊2C2, C25⋊3(C2×C10), (C5×C25)⋊3C22, (C5×C10).7D5, C5.1(D5×C10), C10.4(C5×D5), SmallGroup(500,28)
Series: Derived ►Chief ►Lower central ►Upper central
| C25 — C10×D25 |
Generators and relations for C10×D25
G = < a,b,c | a10=b25=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C10×D25
►On 100 points
Generators in S100
(1 28 11 38 21 48 6 33 16 43)(2 29 12 39 22 49 7 34 17 44)(3 30 13 40 23 50 8 35 18 45)(4 31 14 41 24 26 9 36 19 46)(5 32 15 42 25 27 10 37 20 47)(51 79 66 94 56 84 71 99 61 89)(52 80 67 95 57 85 72 100 62 90)(53 81 68 96 58 86 73 76 63 91)(54 82 69 97 59 87 74 77 64 92)(55 83 70 98 60 88 75 78 65 93)
(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)(2 67)(3 66)(4 65)(5 64)(6 63)(7 62)(8 61)(9 60)(10 59)(11 58)(12 57)(13 56)(14 55)(15 54)(16 53)(17 52)(18 51)(19 75)(20 74)(21 73)(22 72)(23 71)(24 70)(25 69)(26 98)(27 97)(28 96)(29 95)(30 94)(31 93)(32 92)(33 91)(34 90)(35 89)(36 88)(37 87)(38 86)(39 85)(40 84)(41 83)(42 82)(43 81)(44 80)(45 79)(46 78)(47 77)(48 76)(49 100)(50 99)
G:=sub<Sym(100)| (1,28,11,38,21,48,6,33,16,43)(2,29,12,39,22,49,7,34,17,44)(3,30,13,40,23,50,8,35,18,45)(4,31,14,41,24,26,9,36,19,46)(5,32,15,42,25,27,10,37,20,47)(51,79,66,94,56,84,71,99,61,89)(52,80,67,95,57,85,72,100,62,90)(53,81,68,96,58,86,73,76,63,91)(54,82,69,97,59,87,74,77,64,92)(55,83,70,98,60,88,75,78,65,93), (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)(2,67)(3,66)(4,65)(5,64)(6,63)(7,62)(8,61)(9,60)(10,59)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,75)(20,74)(21,73)(22,72)(23,71)(24,70)(25,69)(26,98)(27,97)(28,96)(29,95)(30,94)(31,93)(32,92)(33,91)(34,90)(35,89)(36,88)(37,87)(38,86)(39,85)(40,84)(41,83)(42,82)(43,81)(44,80)(45,79)(46,78)(47,77)(48,76)(49,100)(50,99)>;
G:=Group( (1,28,11,38,21,48,6,33,16,43)(2,29,12,39,22,49,7,34,17,44)(3,30,13,40,23,50,8,35,18,45)(4,31,14,41,24,26,9,36,19,46)(5,32,15,42,25,27,10,37,20,47)(51,79,66,94,56,84,71,99,61,89)(52,80,67,95,57,85,72,100,62,90)(53,81,68,96,58,86,73,76,63,91)(54,82,69,97,59,87,74,77,64,92)(55,83,70,98,60,88,75,78,65,93), (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)(2,67)(3,66)(4,65)(5,64)(6,63)(7,62)(8,61)(9,60)(10,59)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,75)(20,74)(21,73)(22,72)(23,71)(24,70)(25,69)(26,98)(27,97)(28,96)(29,95)(30,94)(31,93)(32,92)(33,91)(34,90)(35,89)(36,88)(37,87)(38,86)(39,85)(40,84)(41,83)(42,82)(43,81)(44,80)(45,79)(46,78)(47,77)(48,76)(49,100)(50,99) );
G=PermutationGroup([[(1,28,11,38,21,48,6,33,16,43),(2,29,12,39,22,49,7,34,17,44),(3,30,13,40,23,50,8,35,18,45),(4,31,14,41,24,26,9,36,19,46),(5,32,15,42,25,27,10,37,20,47),(51,79,66,94,56,84,71,99,61,89),(52,80,67,95,57,85,72,100,62,90),(53,81,68,96,58,86,73,76,63,91),(54,82,69,97,59,87,74,77,64,92),(55,83,70,98,60,88,75,78,65,93)], [(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),(2,67),(3,66),(4,65),(5,64),(6,63),(7,62),(8,61),(9,60),(10,59),(11,58),(12,57),(13,56),(14,55),(15,54),(16,53),(17,52),(18,51),(19,75),(20,74),(21,73),(22,72),(23,71),(24,70),(25,69),(26,98),(27,97),(28,96),(29,95),(30,94),(31,93),(32,92),(33,91),(34,90),(35,89),(36,88),(37,87),(38,86),(39,85),(40,84),(41,83),(42,82),(43,81),(44,80),(45,79),(46,78),(47,77),(48,76),(49,100),(50,99)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 10A | 10B | 10C | 10D | 10E | ··· | 10N | 10O | ··· | 10V | 25A | ··· | 25AX | 50A | ··· | 50AX |
| order | 1 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 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 | C2 | C5 | C10 | C10 | D5 | D10 | D25 | C5×D5 | D50 | D5×C10 | C5×D25 | C10×D25 |
| kernel | C10×D25 | C5×D25 | C5×C50 | D50 | D25 | C50 | C5×C10 | C52 | C10 | C10 | C5 | C5 | C2 | C1 |
| # reps | 1 | 2 | 1 | 4 | 8 | 4 | 2 | 2 | 10 | 8 | 10 | 8 | 40 | 40 |
Matrix representation of C10×D25 ►in GL3(𝔽101) generated by
| 65 | 0 | 0 |
| 0 | 84 | 0 |
| 0 | 0 | 84 |
| 1 | 0 | 0 |
| 0 | 88 | 20 |
| 0 | 0 | 31 |
| 1 | 0 | 0 |
| 0 | 88 | 20 |
| 0 | 32 | 13 |
G:=sub<GL(3,GF(101))| [65,0,0,0,84,0,0,0,84],[1,0,0,0,88,0,0,20,31],[1,0,0,0,88,32,0,20,13] >;
C10×D25 in GAP, Magma, Sage, TeX
C_{10}\times D_{25} % in TeX
G:=Group("C10xD25"); // GroupNames label
G:=SmallGroup(500,28);
// by ID
G=gap.SmallGroup(500,28);
# by ID
G:=PCGroup([5,-2,-2,-5,-5,-5,3603,418,10004]);
// Polycyclic
G:=Group<a,b,c|a^10=b^25=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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