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## G = S3×C45order 270 = 2·33·5

### Direct product of C45 and S3

Aliases: S3×C45, C3⋊C90, C153C18, C32.2C30, (C3×C9)⋊1C10, (C3×C45)⋊1C2, (S3×C15).C3, (C3×S3).C15, C3.4(S3×C15), C15.8(C3×S3), (C3×C15).5C6, SmallGroup(270,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C45
 Chief series C1 — C3 — C32 — C3×C15 — C3×C45 — S3×C45
 Lower central C3 — S3×C45
 Upper central C1 — C45

Generators and relations for S3×C45
G = < a,b,c | a45=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of S3×C45
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 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 82)(2 83)(3 84)(4 85)(5 86)(6 87)(7 88)(8 89)(9 90)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(43 79)(44 80)(45 81)

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

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

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

135 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 5A 5B 5C 5D 6A 6B 9A ··· 9F 9G ··· 9L 10A 10B 10C 10D 15A ··· 15H 15I ··· 15T 18A ··· 18F 30A ··· 30H 45A ··· 45X 45Y ··· 45AV 90A ··· 90X order 1 2 3 3 3 3 3 5 5 5 5 6 6 9 ··· 9 9 ··· 9 10 10 10 10 15 ··· 15 15 ··· 15 18 ··· 18 30 ··· 30 45 ··· 45 45 ··· 45 90 ··· 90 size 1 3 1 1 2 2 2 1 1 1 1 3 3 1 ··· 1 2 ··· 2 3 3 3 3 1 ··· 1 2 ··· 2 3 ··· 3 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

135 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + image C1 C2 C3 C5 C6 C9 C10 C15 C18 C30 C45 C90 S3 C3×S3 C5×S3 S3×C9 S3×C15 S3×C45 kernel S3×C45 C3×C45 S3×C15 S3×C9 C3×C15 C5×S3 C3×C9 C3×S3 C15 C32 S3 C3 C45 C15 C9 C5 C3 C1 # reps 1 1 2 4 2 6 4 8 6 8 24 24 1 2 4 6 8 24

Matrix representation of S3×C45 in GL3(𝔽181) generated by

 126 0 0 0 65 0 0 0 65
,
 1 0 0 0 48 158 0 0 132
,
 180 0 0 0 41 126 0 47 140
G:=sub<GL(3,GF(181))| [126,0,0,0,65,0,0,0,65],[1,0,0,0,48,0,0,158,132],[180,0,0,0,41,47,0,126,140] >;

S3×C45 in GAP, Magma, Sage, TeX

S_3\times C_{45}
% in TeX

G:=Group("S3xC45");
// GroupNames label

G:=SmallGroup(270,9);
// by ID

G=gap.SmallGroup(270,9);
# by ID

G:=PCGroup([5,-2,-3,-5,-3,-3,156,4504]);
// Polycyclic

G:=Group<a,b,c|a^45=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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