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G = C3×D45order 270 = 2·33·5

Direct product of C3 and D45

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

Aliases: C3×D45, C455C6, C152D9, C32.2D15, C5⋊(C3×D9), (C3×C9)⋊2D5, (C3×C45)⋊2C2, C93(C3×D5), C15.1(C3×S3), (C3×C15).4S3, C3.1(C3×D15), SmallGroup(270,12)

Series: Derived Chief Lower central Upper central

C1C45 — C3×D45
C1C3C15C45C3×C45 — C3×D45
C45 — C3×D45
C1C3

Generators and relations for C3×D45
 G = < a,b,c | a3=b45=c2=1, ab=ba, ac=ca, cbc=b-1 >

45C2
2C3
15S3
45C6
2C9
9D5
2C15
5D9
15C3×S3
3D15
9C3×D5
2C45
5C3×D9
3C3×D15

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

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

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

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

72 conjugacy classes

class 1  2 3A3B3C3D3E5A5B6A6B9A···9I15A···15P45A···45AJ
order123333355669···915···1545···45
size145112222245452···22···22···2

72 irreducible representations

dim11112222222222
type+++++++
imageC1C2C3C6S3D5D9C3×S3C3×D5D15C3×D9D45C3×D15C3×D45
kernelC3×D45C3×C45D45C45C3×C15C3×C9C15C15C9C32C5C3C3C1
# reps1122123244612824

Matrix representation of C3×D45 in GL2(𝔽181) generated by

480
048
,
1440
044
,
01
10
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

Subgroup lattice of C3×D45 in TeX

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