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## G = D5×C45order 450 = 2·32·52

### Direct product of C45 and D5

Aliases: D5×C45, C5⋊C90, C453C10, C523C18, 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

 Derived series C1 — C5 — D5×C45
 Chief series C1 — C5 — C15 — C5×C15 — C5×C45 — D5×C45
 Lower central C5 — D5×C45
 Upper central C1 — 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 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(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 46)(29 47)(30 48)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)

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), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)>;

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), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63) );

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)], [(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,71),(9,72),(10,73),(11,74),(12,75),(13,76),(14,77),(15,78),(16,79),(17,80),(18,81),(19,82),(20,83),(21,84),(22,85),(23,86),(24,87),(25,88),(26,89),(27,90),(28,46),(29,47),(30,48),(31,49),(32,50),(33,51),(34,52),(35,53),(36,54),(37,55),(38,56),(39,57),(40,58),(41,59),(42,60),(43,61),(44,62),(45,63)])

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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