direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×D45, C45⋊5C6, C15⋊2D9, C32.2D15, C5⋊(C3×D9), (C3×C9)⋊2D5, (C3×C45)⋊2C2, C9⋊3(C3×D5), C15.1(C3×S3), (C3×C15).4S3, C3.1(C3×D15), SmallGroup(270,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C45 — C3×D45 |
Generators and relations for C3×D45
G = < a,b,c | a3=b45=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D45
►On 90 points
Generators in S90
(1 31 16)(2 32 17)(3 33 18)(4 34 19)(5 35 20)(6 36 21)(7 37 22)(8 38 23)(9 39 24)(10 40 25)(11 41 26)(12 42 27)(13 43 28)(14 44 29)(15 45 30)(46 61 76)(47 62 77)(48 63 78)(49 64 79)(50 65 80)(51 66 81)(52 67 82)(53 68 83)(54 69 84)(55 70 85)(56 71 86)(57 72 87)(58 73 88)(59 74 89)(60 75 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 64)(2 63)(3 62)(4 61)(5 60)(6 59)(7 58)(8 57)(9 56)(10 55)(11 54)(12 53)(13 52)(14 51)(15 50)(16 49)(17 48)(18 47)(19 46)(20 90)(21 89)(22 88)(23 87)(24 86)(25 85)(26 84)(27 83)(28 82)(29 81)(30 80)(31 79)(32 78)(33 77)(34 76)(35 75)(36 74)(37 73)(38 72)(39 71)(40 70)(41 69)(42 68)(43 67)(44 66)(45 65)
G:=sub<Sym(90)| (1,31,16)(2,32,17)(3,33,18)(4,34,19)(5,35,20)(6,36,21)(7,37,22)(8,38,23)(9,39,24)(10,40,25)(11,41,26)(12,42,27)(13,43,28)(14,44,29)(15,45,30)(46,61,76)(47,62,77)(48,63,78)(49,64,79)(50,65,80)(51,66,81)(52,67,82)(53,68,83)(54,69,84)(55,70,85)(56,71,86)(57,72,87)(58,73,88)(59,74,89)(60,75,90), (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), (1,64)(2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,90)(21,89)(22,88)(23,87)(24,86)(25,85)(26,84)(27,83)(28,82)(29,81)(30,80)(31,79)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,70)(41,69)(42,68)(43,67)(44,66)(45,65)>;
G:=Group( (1,31,16)(2,32,17)(3,33,18)(4,34,19)(5,35,20)(6,36,21)(7,37,22)(8,38,23)(9,39,24)(10,40,25)(11,41,26)(12,42,27)(13,43,28)(14,44,29)(15,45,30)(46,61,76)(47,62,77)(48,63,78)(49,64,79)(50,65,80)(51,66,81)(52,67,82)(53,68,83)(54,69,84)(55,70,85)(56,71,86)(57,72,87)(58,73,88)(59,74,89)(60,75,90), (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), (1,64)(2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,90)(21,89)(22,88)(23,87)(24,86)(25,85)(26,84)(27,83)(28,82)(29,81)(30,80)(31,79)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,70)(41,69)(42,68)(43,67)(44,66)(45,65) );
G=PermutationGroup([[(1,31,16),(2,32,17),(3,33,18),(4,34,19),(5,35,20),(6,36,21),(7,37,22),(8,38,23),(9,39,24),(10,40,25),(11,41,26),(12,42,27),(13,43,28),(14,44,29),(15,45,30),(46,61,76),(47,62,77),(48,63,78),(49,64,79),(50,65,80),(51,66,81),(52,67,82),(53,68,83),(54,69,84),(55,70,85),(56,71,86),(57,72,87),(58,73,88),(59,74,89),(60,75,90)], [(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)], [(1,64),(2,63),(3,62),(4,61),(5,60),(6,59),(7,58),(8,57),(9,56),(10,55),(11,54),(12,53),(13,52),(14,51),(15,50),(16,49),(17,48),(18,47),(19,46),(20,90),(21,89),(22,88),(23,87),(24,86),(25,85),(26,84),(27,83),(28,82),(29,81),(30,80),(31,79),(32,78),(33,77),(34,76),(35,75),(36,74),(37,73),(38,72),(39,71),(40,70),(41,69),(42,68),(43,67),(44,66),(45,65)]])
72 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 5A | 5B | 6A | 6B | 9A | ··· | 9I | 15A | ··· | 15P | 45A | ··· | 45AJ |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 5 | 5 | 6 | 6 | 9 | ··· | 9 | 15 | ··· | 15 | 45 | ··· | 45 |
| size | 1 | 45 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 45 | 45 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C3 | C6 | S3 | D5 | D9 | C3×S3 | C3×D5 | D15 | C3×D9 | D45 | C3×D15 | C3×D45 |
| kernel | C3×D45 | C3×C45 | D45 | C45 | C3×C15 | C3×C9 | C15 | C15 | C9 | C32 | C5 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 2 | 3 | 2 | 4 | 4 | 6 | 12 | 8 | 24 |
Matrix representation of C3×D45 ►in GL2(𝔽181) generated by
| 48 | 0 |
| 0 | 48 |
| 144 | 0 |
| 0 | 44 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(181))| [48,0,0,48],[144,0,0,44],[0,1,1,0] >;
C3×D45 in GAP, Magma, Sage, TeX
C_3\times D_{45} % in TeX
G:=Group("C3xD45"); // GroupNames label
G:=SmallGroup(270,12);
// by ID
G=gap.SmallGroup(270,12);
# by ID
G:=PCGroup([5,-2,-3,-3,-5,-3,1532,462,1443,4504]);
// Polycyclic
G:=Group<a,b,c|a^3=b^45=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export