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## G = C15×D9order 270 = 2·33·5

### Direct product of C15 and D9

Aliases: C15×D9, C456C6, C93C30, (C3×C9)⋊2C10, (C3×C45)⋊4C2, C3.1(S3×C15), C15.5(C3×S3), (C3×C15).5S3, C32.2(C5×S3), SmallGroup(270,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C15×D9
 Chief series C1 — C3 — C9 — C45 — C3×C45 — C15×D9
 Lower central C9 — C15×D9
 Upper central C1 — C15

Generators and relations for C15×D9
G = < a,b,c | a15=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C15×D9
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 31 11 29 41 6 24 36)(2 20 32 12 30 42 7 25 37)(3 21 33 13 16 43 8 26 38)(4 22 34 14 17 44 9 27 39)(5 23 35 15 18 45 10 28 40)(46 65 83 51 70 88 56 75 78)(47 66 84 52 71 89 57 61 79)(48 67 85 53 72 90 58 62 80)(49 68 86 54 73 76 59 63 81)(50 69 87 55 74 77 60 64 82)
(1 70)(2 71)(3 72)(4 73)(5 74)(6 75)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 48)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 46)(30 47)(31 83)(32 84)(33 85)(34 86)(35 87)(36 88)(37 89)(38 90)(39 76)(40 77)(41 78)(42 79)(43 80)(44 81)(45 82)

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

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

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

90 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 5A 5B 5C 5D 6A 6B 9A ··· 9I 10A 10B 10C 10D 15A ··· 15H 15I ··· 15T 30A ··· 30H 45A ··· 45AJ order 1 2 3 3 3 3 3 5 5 5 5 6 6 9 ··· 9 10 10 10 10 15 ··· 15 15 ··· 15 30 ··· 30 45 ··· 45 size 1 9 1 1 2 2 2 1 1 1 1 9 9 2 ··· 2 9 9 9 9 1 ··· 1 2 ··· 2 9 ··· 9 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + image C1 C2 C3 C5 C6 C10 C15 C30 S3 D9 C3×S3 C5×S3 C3×D9 C5×D9 S3×C15 C15×D9 kernel C15×D9 C3×C45 C5×D9 C3×D9 C45 C3×C9 D9 C9 C3×C15 C15 C15 C32 C5 C3 C3 C1 # reps 1 1 2 4 2 4 8 8 1 3 2 4 6 12 8 24

Matrix representation of C15×D9 in GL3(𝔽181) generated by

 27 0 0 0 132 0 0 0 132
,
 1 0 0 0 80 0 0 56 43
,
 1 0 0 0 72 30 0 147 109
G:=sub<GL(3,GF(181))| [27,0,0,0,132,0,0,0,132],[1,0,0,0,80,56,0,0,43],[1,0,0,0,72,147,0,30,109] >;

C15×D9 in GAP, Magma, Sage, TeX

C_{15}\times D_9
% in TeX

G:=Group("C15xD9");
// GroupNames label

G:=SmallGroup(270,8);
// by ID

G=gap.SmallGroup(270,8);
# by ID

G:=PCGroup([5,-2,-3,-5,-3,-3,3003,138,4504]);
// Polycyclic

G:=Group<a,b,c|a^15=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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