direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C5×D45, C45⋊2D5, C45⋊1C10, C52⋊2D9, C15.5D15, C9⋊(C5×D5), C5⋊(C5×D9), (C5×C45)⋊2C2, C3.(C5×D15), C15.1(C5×S3), (C5×C15).3S3, SmallGroup(450,17)
Series: Derived ►Chief ►Lower central ►Upper central
| C45 — C5×D45 |
Generators and relations for C5×D45
G = < a,b,c | a5=b45=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C5×D45
►On 90 points
Generators in S90
(1 19 37 10 28)(2 20 38 11 29)(3 21 39 12 30)(4 22 40 13 31)(5 23 41 14 32)(6 24 42 15 33)(7 25 43 16 34)(8 26 44 17 35)(9 27 45 18 36)(46 73 55 82 64)(47 74 56 83 65)(48 75 57 84 66)(49 76 58 85 67)(50 77 59 86 68)(51 78 60 87 69)(52 79 61 88 70)(53 80 62 89 71)(54 81 63 90 72)
(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 63)(2 62)(3 61)(4 60)(5 59)(6 58)(7 57)(8 56)(9 55)(10 54)(11 53)(12 52)(13 51)(14 50)(15 49)(16 48)(17 47)(18 46)(19 90)(20 89)(21 88)(22 87)(23 86)(24 85)(25 84)(26 83)(27 82)(28 81)(29 80)(30 79)(31 78)(32 77)(33 76)(34 75)(35 74)(36 73)(37 72)(38 71)(39 70)(40 69)(41 68)(42 67)(43 66)(44 65)(45 64)
G:=sub<Sym(90)| (1,19,37,10,28)(2,20,38,11,29)(3,21,39,12,30)(4,22,40,13,31)(5,23,41,14,32)(6,24,42,15,33)(7,25,43,16,34)(8,26,44,17,35)(9,27,45,18,36)(46,73,55,82,64)(47,74,56,83,65)(48,75,57,84,66)(49,76,58,85,67)(50,77,59,86,68)(51,78,60,87,69)(52,79,61,88,70)(53,80,62,89,71)(54,81,63,90,72), (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), (1,63)(2,62)(3,61)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,54)(11,53)(12,52)(13,51)(14,50)(15,49)(16,48)(17,47)(18,46)(19,90)(20,89)(21,88)(22,87)(23,86)(24,85)(25,84)(26,83)(27,82)(28,81)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,72)(38,71)(39,70)(40,69)(41,68)(42,67)(43,66)(44,65)(45,64)>;
G:=Group( (1,19,37,10,28)(2,20,38,11,29)(3,21,39,12,30)(4,22,40,13,31)(5,23,41,14,32)(6,24,42,15,33)(7,25,43,16,34)(8,26,44,17,35)(9,27,45,18,36)(46,73,55,82,64)(47,74,56,83,65)(48,75,57,84,66)(49,76,58,85,67)(50,77,59,86,68)(51,78,60,87,69)(52,79,61,88,70)(53,80,62,89,71)(54,81,63,90,72), (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), (1,63)(2,62)(3,61)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,54)(11,53)(12,52)(13,51)(14,50)(15,49)(16,48)(17,47)(18,46)(19,90)(20,89)(21,88)(22,87)(23,86)(24,85)(25,84)(26,83)(27,82)(28,81)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,72)(38,71)(39,70)(40,69)(41,68)(42,67)(43,66)(44,65)(45,64) );
G=PermutationGroup([[(1,19,37,10,28),(2,20,38,11,29),(3,21,39,12,30),(4,22,40,13,31),(5,23,41,14,32),(6,24,42,15,33),(7,25,43,16,34),(8,26,44,17,35),(9,27,45,18,36),(46,73,55,82,64),(47,74,56,83,65),(48,75,57,84,66),(49,76,58,85,67),(50,77,59,86,68),(51,78,60,87,69),(52,79,61,88,70),(53,80,62,89,71),(54,81,63,90,72)], [(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)], [(1,63),(2,62),(3,61),(4,60),(5,59),(6,58),(7,57),(8,56),(9,55),(10,54),(11,53),(12,52),(13,51),(14,50),(15,49),(16,48),(17,47),(18,46),(19,90),(20,89),(21,88),(22,87),(23,86),(24,85),(25,84),(26,83),(27,82),(28,81),(29,80),(30,79),(31,78),(32,77),(33,76),(34,75),(35,74),(36,73),(37,72),(38,71),(39,70),(40,69),(41,68),(42,67),(43,66),(44,65),(45,64)]])
120 conjugacy classes
| class | 1 | 2 | 3 | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 9A | 9B | 9C | 10A | 10B | 10C | 10D | 15A | ··· | 15X | 45A | ··· | 45BT |
| order | 1 | 2 | 3 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 9 | 9 | 9 | 10 | 10 | 10 | 10 | 15 | ··· | 15 | 45 | ··· | 45 |
| size | 1 | 45 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 45 | 45 | 45 | 45 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C5 | C10 | S3 | D5 | D9 | C5×S3 | D15 | C5×D5 | C5×D9 | D45 | C5×D15 | C5×D45 |
| kernel | C5×D45 | C5×C45 | D45 | C45 | C5×C15 | C45 | C52 | C15 | C15 | C9 | C5 | C5 | C3 | C1 |
| # reps | 1 | 1 | 4 | 4 | 1 | 2 | 3 | 4 | 4 | 8 | 12 | 12 | 16 | 48 |
Matrix representation of C5×D45 ►in GL2(𝔽181) generated by
| 42 | 0 |
| 0 | 42 |
| 16 | 0 |
| 0 | 34 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(181))| [42,0,0,42],[16,0,0,34],[0,1,1,0] >;
C5×D45 in GAP, Magma, Sage, TeX
C_5\times D_{45} % in TeX
G:=Group("C5xD45"); // GroupNames label
G:=SmallGroup(450,17);
// by ID
G=gap.SmallGroup(450,17);
# by ID
G:=PCGroup([5,-2,-5,-3,-5,-3,2552,462,2403,7504]);
// Polycyclic
G:=Group<a,b,c|a^5=b^45=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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