direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C10×D9, C18⋊C10, C90⋊2C2, C30.6S3, C45⋊3C22, C15.3D6, C9⋊(C2×C10), C3.(S3×C10), C6.2(C5×S3), SmallGroup(180,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C10×D9 |
Generators and relations for C10×D9
G = < a,b,c | a10=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C10×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 27 34 13 83 53 79 67 41)(2 28 35 14 84 54 80 68 42)(3 29 36 15 85 55 71 69 43)(4 30 37 16 86 56 72 70 44)(5 21 38 17 87 57 73 61 45)(6 22 39 18 88 58 74 62 46)(7 23 40 19 89 59 75 63 47)(8 24 31 20 90 60 76 64 48)(9 25 32 11 81 51 77 65 49)(10 26 33 12 82 52 78 66 50)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 76)(32 77)(33 78)(34 79)(35 80)(36 71)(37 72)(38 73)(39 74)(40 75)
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), (1,27,34,13,83,53,79,67,41)(2,28,35,14,84,54,80,68,42)(3,29,36,15,85,55,71,69,43)(4,30,37,16,86,56,72,70,44)(5,21,38,17,87,57,73,61,45)(6,22,39,18,88,58,74,62,46)(7,23,40,19,89,59,75,63,47)(8,24,31,20,90,60,76,64,48)(9,25,32,11,81,51,77,65,49)(10,26,33,12,82,52,78,66,50), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,76)(32,77)(33,78)(34,79)(35,80)(36,71)(37,72)(38,73)(39,74)(40,75)>;
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), (1,27,34,13,83,53,79,67,41)(2,28,35,14,84,54,80,68,42)(3,29,36,15,85,55,71,69,43)(4,30,37,16,86,56,72,70,44)(5,21,38,17,87,57,73,61,45)(6,22,39,18,88,58,74,62,46)(7,23,40,19,89,59,75,63,47)(8,24,31,20,90,60,76,64,48)(9,25,32,11,81,51,77,65,49)(10,26,33,12,82,52,78,66,50), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,76)(32,77)(33,78)(34,79)(35,80)(36,71)(37,72)(38,73)(39,74)(40,75) );
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)], [(1,27,34,13,83,53,79,67,41),(2,28,35,14,84,54,80,68,42),(3,29,36,15,85,55,71,69,43),(4,30,37,16,86,56,72,70,44),(5,21,38,17,87,57,73,61,45),(6,22,39,18,88,58,74,62,46),(7,23,40,19,89,59,75,63,47),(8,24,31,20,90,60,76,64,48),(9,25,32,11,81,51,77,65,49),(10,26,33,12,82,52,78,66,50)], [(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,76),(32,77),(33,78),(34,79),(35,80),(36,71),(37,72),(38,73),(39,74),(40,75)]])
C10×D9 is a maximal subgroup of
C5⋊D36 C45⋊D4
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 5A | 5B | 5C | 5D | 6 | 9A | 9B | 9C | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 15A | 15B | 15C | 15D | 18A | 18B | 18C | 30A | 30B | 30C | 30D | 45A | ··· | 45L | 90A | ··· | 90L |
| order | 1 | 2 | 2 | 2 | 3 | 5 | 5 | 5 | 5 | 6 | 9 | 9 | 9 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 18 | 18 | 18 | 30 | 30 | 30 | 30 | 45 | ··· | 45 | 90 | ··· | 90 |
| size | 1 | 1 | 9 | 9 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 9 | ··· | 9 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
60 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 | S3 | D6 | D9 | C5×S3 | D18 | S3×C10 | C5×D9 | C10×D9 |
| kernel | C10×D9 | C5×D9 | C90 | D18 | D9 | C18 | C30 | C15 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 4 | 8 | 4 | 1 | 1 | 3 | 4 | 3 | 4 | 12 | 12 |
Matrix representation of C10×D9 ►in GL3(𝔽181) generated by
| 56 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 127 | 131 |
| 0 | 50 | 177 |
| 180 | 0 | 0 |
| 0 | 127 | 131 |
| 0 | 4 | 54 |
G:=sub<GL(3,GF(181))| [56,0,0,0,1,0,0,0,1],[1,0,0,0,127,50,0,131,177],[180,0,0,0,127,4,0,131,54] >;
C10×D9 in GAP, Magma, Sage, TeX
C_{10}\times D_9 % in TeX
G:=Group("C10xD9"); // GroupNames label
G:=SmallGroup(180,10);
// by ID
G=gap.SmallGroup(180,10);
# by ID
G:=PCGroup([5,-2,-2,-5,-3,-3,2003,138,3004]);
// Polycyclic
G:=Group<a,b,c|a^10=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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