Copied to
clipboard

## G = C10×D9order 180 = 22·32·5

### Direct product of C10 and D9

Aliases: C10×D9, C18⋊C10, C902C2, C30.6S3, C453C22, C15.3D6, C9⋊(C2×C10), C3.(S3×C10), C6.2(C5×S3), SmallGroup(180,10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C10×D9
 Chief series C1 — C3 — C9 — C45 — C5×D9 — C10×D9
 Lower central C9 — C10×D9
 Upper central C1 — C10

Generators and relations for C10×D9
G = < a,b,c | a10=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C10×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 54 80 86 38 17 42 67 23)(2 55 71 87 39 18 43 68 24)(3 56 72 88 40 19 44 69 25)(4 57 73 89 31 20 45 70 26)(5 58 74 90 32 11 46 61 27)(6 59 75 81 33 12 47 62 28)(7 60 76 82 34 13 48 63 29)(8 51 77 83 35 14 49 64 30)(9 52 78 84 36 15 50 65 21)(10 53 79 85 37 16 41 66 22)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 21)(10 22)(11 90)(12 81)(13 82)(14 83)(15 84)(16 85)(17 86)(18 87)(19 88)(20 89)(41 79)(42 80)(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 77)(50 78)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)(57 70)(58 61)(59 62)(60 63)

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

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

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

C10×D9 is a maximal subgroup of   C5⋊D36  C45⋊D4

60 conjugacy classes

 class 1 2A 2B 2C 3 5A 5B 5C 5D 6 9A 9B 9C 10A 10B 10C 10D 10E ··· 10L 15A 15B 15C 15D 18A 18B 18C 30A 30B 30C 30D 45A ··· 45L 90A ··· 90L order 1 2 2 2 3 5 5 5 5 6 9 9 9 10 10 10 10 10 ··· 10 15 15 15 15 18 18 18 30 30 30 30 45 ··· 45 90 ··· 90 size 1 1 9 9 2 1 1 1 1 2 2 2 2 1 1 1 1 9 ··· 9 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C5 C10 C10 S3 D6 D9 C5×S3 D18 S3×C10 C5×D9 C10×D9 kernel C10×D9 C5×D9 C90 D18 D9 C18 C30 C15 C10 C6 C5 C3 C2 C1 # reps 1 2 1 4 8 4 1 1 3 4 3 4 12 12

Matrix representation of C10×D9 in GL3(𝔽181) generated by

 56 0 0 0 1 0 0 0 1
,
 1 0 0 0 127 131 0 50 177
,
 180 0 0 0 127 131 0 4 54
G:=sub<GL(3,GF(181))| [56,0,0,0,1,0,0,0,1],[1,0,0,0,127,50,0,131,177],[180,0,0,0,127,4,0,131,54] >;

C10×D9 in GAP, Magma, Sage, TeX

C_{10}\times D_9
% in TeX

G:=Group("C10xD9");
// GroupNames label

G:=SmallGroup(180,10);
// by ID

G=gap.SmallGroup(180,10);
# by ID

G:=PCGroup([5,-2,-2,-5,-3,-3,2003,138,3004]);
// Polycyclic

G:=Group<a,b,c|a^10=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

׿
×
𝔽