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## G = C9×D15order 270 = 2·33·5

### Direct product of C9 and D15

Aliases: C9×D15, C453S3, C151C18, C5⋊(S3×C9), C3⋊(C9×D5), (C3×C9)⋊1D5, (C3×C45)⋊7C2, (C3×D15).C3, C15.4(C3×S3), (C3×C15).3C6, C3.4(C3×D15), C32.2(C3×D5), SmallGroup(270,13)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C9×D15
 Chief series C1 — C5 — C15 — C3×C15 — C3×C45 — C9×D15
 Lower central C15 — C9×D15
 Upper central C1 — C9

Generators and relations for C9×D15
G = < a,b,c | a9=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C9×D15
On 90 points
Generators in S90
(1 37 21 6 42 26 11 32 16)(2 38 22 7 43 27 12 33 17)(3 39 23 8 44 28 13 34 18)(4 40 24 9 45 29 14 35 19)(5 41 25 10 31 30 15 36 20)(46 82 66 56 77 61 51 87 71)(47 83 67 57 78 62 52 88 72)(48 84 68 58 79 63 53 89 73)(49 85 69 59 80 64 54 90 74)(50 86 70 60 81 65 55 76 75)
(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 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 60)(9 59)(10 58)(11 57)(12 56)(13 55)(14 54)(15 53)(16 62)(17 61)(18 75)(19 74)(20 73)(21 72)(22 71)(23 70)(24 69)(25 68)(26 67)(27 66)(28 65)(29 64)(30 63)(31 79)(32 78)(33 77)(34 76)(35 90)(36 89)(37 88)(38 87)(39 86)(40 85)(41 84)(42 83)(43 82)(44 81)(45 80)

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

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

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

81 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 5A 5B 6A 6B 9A ··· 9F 9G ··· 9L 15A ··· 15P 18A ··· 18F 45A ··· 45AJ order 1 2 3 3 3 3 3 5 5 6 6 9 ··· 9 9 ··· 9 15 ··· 15 18 ··· 18 45 ··· 45 size 1 15 1 1 2 2 2 2 2 15 15 1 ··· 1 2 ··· 2 2 ··· 2 15 ··· 15 2 ··· 2

81 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + image C1 C2 C3 C6 C9 C18 S3 D5 C3×S3 D15 C3×D5 S3×C9 C9×D5 C3×D15 C9×D15 kernel C9×D15 C3×C45 C3×D15 C3×C15 D15 C15 C45 C3×C9 C15 C9 C32 C5 C3 C3 C1 # reps 1 1 2 2 6 6 1 2 2 4 4 6 12 8 24

Matrix representation of C9×D15 in GL2(𝔽181) generated by

 62 0 0 62
,
 25 0 0 29
,
 0 29 25 0
G:=sub<GL(2,GF(181))| [62,0,0,62],[25,0,0,29],[0,25,29,0] >;

C9×D15 in GAP, Magma, Sage, TeX

C_9\times D_{15}
% in TeX

G:=Group("C9xD15");
// GroupNames label

G:=SmallGroup(270,13);
// by ID

G=gap.SmallGroup(270,13);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-5,36,723,5404]);
// Polycyclic

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

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