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G = C5×D45order 450 = 2·32·52

Direct product of C5 and D45

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C5×D45, C452D5, C451C10, C522D9, 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

C1C45 — C5×D45
C1C3C15C45C5×C45 — C5×D45
C45 — C5×D45
C1C5

Generators and relations for C5×D45
 G = < a,b,c | a5=b45=c2=1, ab=ba, ac=ca, cbc=b-1 >

45C2
2C5
2C5
15S3
9D5
45C10
2C15
2C15
5D9
3D15
15C5×S3
2C45
2C45
9C5×D5
5C5×D9
3C5×D15

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 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)| (1,19,37,10,28)(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), (1,64)(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( (1,19,37,10,28)(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), (1,64)(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([(1,19,37,10,28),(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)], [(1,64),(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)])

120 conjugacy classes

class 1  2  3 5A5B5C5D5E···5N9A9B9C10A10B10C10D15A···15X45A···45BT
order12355555···59991010101015···1545···45
size145211112···2222454545452···22···2

120 irreducible representations

dim11112222222222
type+++++++
imageC1C2C5C10S3D5D9C5×S3D15C5×D5C5×D9D45C5×D15C5×D45
kernelC5×D45C5×C45D45C45C5×C15C45C52C15C15C9C5C5C3C1
# reps114412344812121648

Matrix representation of C5×D45 in GL2(𝔽181) generated by

420
042
,
160
034
,
01
10
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

Export

Subgroup lattice of C5×D45 in TeX

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