direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C15×D9, C45⋊6C6, C9⋊3C30, (C3×C9)⋊2C10, (C3×C45)⋊4C2, C3.1(S3×C15), C15.5(C3×S3), (C3×C15).5S3, C32.2(C5×S3), SmallGroup(270,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C15×D9 |
Generators and relations for C15×D9
G = < a,b,c | a15=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C15×D9
►On 90 points
Generators in S90
(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)
(1 50 20 11 60 30 6 55 25)(2 51 21 12 46 16 7 56 26)(3 52 22 13 47 17 8 57 27)(4 53 23 14 48 18 9 58 28)(5 54 24 15 49 19 10 59 29)(31 79 66 36 84 71 41 89 61)(32 80 67 37 85 72 42 90 62)(33 81 68 38 86 73 43 76 63)(34 82 69 39 87 74 44 77 64)(35 83 70 40 88 75 45 78 65)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 85)(23 86)(24 87)(25 88)(26 89)(27 90)(28 76)(29 77)(30 78)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 61)(57 62)(58 63)(59 64)(60 65)
G:=sub<Sym(90)| (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), (1,50,20,11,60,30,6,55,25)(2,51,21,12,46,16,7,56,26)(3,52,22,13,47,17,8,57,27)(4,53,23,14,48,18,9,58,28)(5,54,24,15,49,19,10,59,29)(31,79,66,36,84,71,41,89,61)(32,80,67,37,85,72,42,90,62)(33,81,68,38,86,73,43,76,63)(34,82,69,39,87,74,44,77,64)(35,83,70,40,88,75,45,78,65), (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,76)(29,77)(30,78)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,61)(57,62)(58,63)(59,64)(60,65)>;
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), (1,50,20,11,60,30,6,55,25)(2,51,21,12,46,16,7,56,26)(3,52,22,13,47,17,8,57,27)(4,53,23,14,48,18,9,58,28)(5,54,24,15,49,19,10,59,29)(31,79,66,36,84,71,41,89,61)(32,80,67,37,85,72,42,90,62)(33,81,68,38,86,73,43,76,63)(34,82,69,39,87,74,44,77,64)(35,83,70,40,88,75,45,78,65), (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,76)(29,77)(30,78)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,61)(57,62)(58,63)(59,64)(60,65) );
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)], [(1,50,20,11,60,30,6,55,25),(2,51,21,12,46,16,7,56,26),(3,52,22,13,47,17,8,57,27),(4,53,23,14,48,18,9,58,28),(5,54,24,15,49,19,10,59,29),(31,79,66,36,84,71,41,89,61),(32,80,67,37,85,72,42,90,62),(33,81,68,38,86,73,43,76,63),(34,82,69,39,87,74,44,77,64),(35,83,70,40,88,75,45,78,65)], [(1,40),(2,41),(3,42),(4,43),(5,44),(6,45),(7,31),(8,32),(9,33),(10,34),(11,35),(12,36),(13,37),(14,38),(15,39),(16,79),(17,80),(18,81),(19,82),(20,83),(21,84),(22,85),(23,86),(24,87),(25,88),(26,89),(27,90),(28,76),(29,77),(30,78),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,61),(57,62),(58,63),(59,64),(60,65)]])
90 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 5A | 5B | 5C | 5D | 6A | 6B | 9A | ··· | 9I | 10A | 10B | 10C | 10D | 15A | ··· | 15H | 15I | ··· | 15T | 30A | ··· | 30H | 45A | ··· | 45AJ |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 5 | 5 | 5 | 5 | 6 | 6 | 9 | ··· | 9 | 10 | 10 | 10 | 10 | 15 | ··· | 15 | 15 | ··· | 15 | 30 | ··· | 30 | 45 | ··· | 45 |
| size | 1 | 9 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 9 | 9 | 2 | ··· | 2 | 9 | 9 | 9 | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 | S3 | D9 | C3×S3 | C5×S3 | C3×D9 | C5×D9 | S3×C15 | C15×D9 |
| kernel | C15×D9 | C3×C45 | C5×D9 | C3×D9 | C45 | C3×C9 | D9 | C9 | C3×C15 | C15 | C15 | C32 | C5 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 4 | 2 | 4 | 8 | 8 | 1 | 3 | 2 | 4 | 6 | 12 | 8 | 24 |
Matrix representation of C15×D9 ►in GL3(𝔽181) generated by
| 27 | 0 | 0 |
| 0 | 132 | 0 |
| 0 | 0 | 132 |
| 1 | 0 | 0 |
| 0 | 80 | 0 |
| 0 | 56 | 43 |
| 1 | 0 | 0 |
| 0 | 72 | 30 |
| 0 | 147 | 109 |
G:=sub<GL(3,GF(181))| [27,0,0,0,132,0,0,0,132],[1,0,0,0,80,56,0,0,43],[1,0,0,0,72,147,0,30,109] >;
C15×D9 in GAP, Magma, Sage, TeX
C_{15}\times D_9 % in TeX
G:=Group("C15xD9"); // GroupNames label
G:=SmallGroup(270,8);
// by ID
G=gap.SmallGroup(270,8);
# by ID
G:=PCGroup([5,-2,-3,-5,-3,-3,3003,138,4504]);
// Polycyclic
G:=Group<a,b,c|a^15=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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