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G = C3xD45order 270 = 2·33·5

Direct product of C3 and D45

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

Aliases: C3xD45, C45:5C6, C15:2D9, C32.2D15, C5:(C3xD9), (C3xC9):2D5, (C3xC45):2C2, C9:3(C3xD5), C15.1(C3xS3), (C3xC15).4S3, C3.1(C3xD15), SmallGroup(270,12)

Series: Derived Chief Lower central Upper central

C1C45 — C3xD45
C1C3C15C45C3xC45 — C3xD45
C45 — C3xD45
C1C3

Generators and relations for C3xD45
 G = < a,b,c | a3=b45=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 176 in 28 conjugacy classes, 14 normal (all characteristic)
Quotients: C1, C2, C3, S3, C6, D5, D9, C3xS3, C3xD5, D15, C3xD9, D45, C3xD15, C3xD45
45C2
2C3
15S3
45C6
2C9
9D5
2C15
5D9
15C3xS3
3D15
9C3xD5
2C45
5C3xD9
3C3xD15

Smallest permutation representation of C3xD45
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 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)| (1,31,16)(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), (1,64)(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( (1,31,16)(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), (1,64)(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([[(1,31,16),(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)], [(1,64),(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)]])

72 conjugacy classes

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

72 irreducible representations

dim11112222222222
type+++++++
imageC1C2C3C6S3D5D9C3xS3C3xD5D15C3xD9D45C3xD15C3xD45
kernelC3xD45C3xC45D45C45C3xC15C3xC9C15C15C9C32C5C3C3C1
# reps1122123244612824

Matrix representation of C3xD45 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] >;

C3xD45 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 C3xD45 in TeX

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