direct product, metacyclic, supersoluble, monomial, A-group
Aliases: D5×C45, C5⋊C90, C45⋊3C10, C52⋊3C18, C15.1C30, (C5×C45)⋊4C2, C3.(D5×C15), (D5×C15).C3, (C3×D5).C15, (C5×C15).3C6, C15.5(C3×D5), SmallGroup(450,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C45 |
Generators and relations for D5×C45
G = < a,b,c | a45=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C45
►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 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 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 71)(10 72)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 79)(18 80)(19 81)(20 82)(21 83)(22 84)(23 85)(24 86)(25 87)(26 88)(27 89)(28 90)(29 46)(30 47)(31 48)(32 49)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 57)(41 58)(42 59)(43 60)(44 61)(45 62)
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), (1,19,37,10,28)(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), (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,71)(10,72)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,79)(18,80)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(25,87)(26,88)(27,89)(28,90)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,57)(41,58)(42,59)(43,60)(44,61)(45,62)>;
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), (1,19,37,10,28)(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), (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,71)(10,72)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,79)(18,80)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(25,87)(26,88)(27,89)(28,90)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,57)(41,58)(42,59)(43,60)(44,61)(45,62) );
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)], [(1,19,37,10,28),(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)], [(1,63),(2,64),(3,65),(4,66),(5,67),(6,68),(7,69),(8,70),(9,71),(10,72),(11,73),(12,74),(13,75),(14,76),(15,77),(16,78),(17,79),(18,80),(19,81),(20,82),(21,83),(22,84),(23,85),(24,86),(25,87),(26,88),(27,89),(28,90),(29,46),(30,47),(31,48),(32,49),(33,50),(34,51),(35,52),(36,53),(37,54),(38,55),(39,56),(40,57),(41,58),(42,59),(43,60),(44,61),(45,62)]])
180 conjugacy classes
| class | 1 | 2 | 3A | 3B | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 6A | 6B | 9A | ··· | 9F | 10A | 10B | 10C | 10D | 15A | ··· | 15H | 15I | ··· | 15AB | 18A | ··· | 18F | 30A | ··· | 30H | 45A | ··· | 45X | 45Y | ··· | 45CF | 90A | ··· | 90X |
| order | 1 | 2 | 3 | 3 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 6 | 6 | 9 | ··· | 9 | 10 | 10 | 10 | 10 | 15 | ··· | 15 | 15 | ··· | 15 | 18 | ··· | 18 | 30 | ··· | 30 | 45 | ··· | 45 | 45 | ··· | 45 | 90 | ··· | 90 |
| size | 1 | 5 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 5 | 5 | 1 | ··· | 1 | 5 | 5 | 5 | 5 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 | 5 | ··· | 5 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||||||||
| image | C1 | C2 | C3 | C5 | C6 | C9 | C10 | C15 | C18 | C30 | C45 | C90 | D5 | C3×D5 | C5×D5 | C9×D5 | D5×C15 | D5×C45 |
| kernel | D5×C45 | C5×C45 | D5×C15 | C9×D5 | C5×C15 | C5×D5 | C45 | C3×D5 | C52 | C15 | D5 | C5 | C45 | C15 | C9 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 4 | 2 | 6 | 4 | 8 | 6 | 8 | 24 | 24 | 2 | 4 | 8 | 12 | 16 | 48 |
Matrix representation of D5×C45 ►in GL3(𝔽181) generated by
| 34 | 0 | 0 |
| 0 | 29 | 0 |
| 0 | 0 | 29 |
| 1 | 0 | 0 |
| 0 | 135 | 0 |
| 0 | 51 | 59 |
| 1 | 0 | 0 |
| 0 | 158 | 123 |
| 0 | 84 | 23 |
G:=sub<GL(3,GF(181))| [34,0,0,0,29,0,0,0,29],[1,0,0,0,135,51,0,0,59],[1,0,0,0,158,84,0,123,23] >;
D5×C45 in GAP, Magma, Sage, TeX
D_5\times C_{45} % in TeX
G:=Group("D5xC45"); // GroupNames label
G:=SmallGroup(450,14);
// by ID
G=gap.SmallGroup(450,14);
# by ID
G:=PCGroup([5,-2,-3,-5,-3,-5,156,9004]);
// Polycyclic
G:=Group<a,b,c|a^45=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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