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

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

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

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

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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