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## G = C3×C9⋊C18order 486 = 2·35

### Direct product of C3 and C9⋊C18

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C3×C9⋊C18
 Chief series C1 — C3 — C9 — C3×C9 — C32×C9 — C3×C9⋊C9 — C3×C9⋊C18
 Lower central C9 — C3×C9⋊C18
 Upper central C1 — C32

Generators and relations for C3×C9⋊C18
G = < a,b,c | a3=b9=c18=1, ab=ba, ac=ca, cbc-1=b2 >

Subgroups: 256 in 90 conjugacy classes, 36 normal (15 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C18, C3×S3, C3×C6, C3×C9, C3×C9, C3×C9, C33, C3×D9, C3×D9, S3×C9, C3×C18, S3×C32, C9⋊C9, C9⋊C9, C32×C9, C32×C9, C9⋊C18, C32×D9, S3×C3×C9, C3×C9⋊C9, C3×C9⋊C18
Quotients: C1, C2, C3, S3, C6, C9, C32, C18, C3×S3, C3×C6, C3×C9, S3×C9, C9⋊C6, C3×C18, S3×C32, C9⋊C18, S3×C3×C9, C3×C9⋊C6, C3×C9⋊C18

Smallest permutation representation of C3×C9⋊C18
On 54 points
Generators in S54
(1 21 42)(2 22 43)(3 23 44)(4 24 45)(5 25 46)(6 26 47)(7 27 48)(8 28 49)(9 29 50)(10 30 51)(11 31 52)(12 32 53)(13 33 54)(14 34 37)(15 35 38)(16 36 39)(17 19 40)(18 20 41)
(1 46 35 13 40 29 7 52 23)(2 36 41 8 24 47 14 30 53)(3 42 25 15 54 19 9 48 31)(4 26 37 10 32 43 16 20 49)(5 38 33 17 50 27 11 44 21)(6 34 51 12 22 39 18 28 45)
(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)

G:=sub<Sym(54)| (1,21,42)(2,22,43)(3,23,44)(4,24,45)(5,25,46)(6,26,47)(7,27,48)(8,28,49)(9,29,50)(10,30,51)(11,31,52)(12,32,53)(13,33,54)(14,34,37)(15,35,38)(16,36,39)(17,19,40)(18,20,41), (1,46,35,13,40,29,7,52,23)(2,36,41,8,24,47,14,30,53)(3,42,25,15,54,19,9,48,31)(4,26,37,10,32,43,16,20,49)(5,38,33,17,50,27,11,44,21)(6,34,51,12,22,39,18,28,45), (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)>;

G:=Group( (1,21,42)(2,22,43)(3,23,44)(4,24,45)(5,25,46)(6,26,47)(7,27,48)(8,28,49)(9,29,50)(10,30,51)(11,31,52)(12,32,53)(13,33,54)(14,34,37)(15,35,38)(16,36,39)(17,19,40)(18,20,41), (1,46,35,13,40,29,7,52,23)(2,36,41,8,24,47,14,30,53)(3,42,25,15,54,19,9,48,31)(4,26,37,10,32,43,16,20,49)(5,38,33,17,50,27,11,44,21)(6,34,51,12,22,39,18,28,45), (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) );

G=PermutationGroup([[(1,21,42),(2,22,43),(3,23,44),(4,24,45),(5,25,46),(6,26,47),(7,27,48),(8,28,49),(9,29,50),(10,30,51),(11,31,52),(12,32,53),(13,33,54),(14,34,37),(15,35,38),(16,36,39),(17,19,40),(18,20,41)], [(1,46,35,13,40,29,7,52,23),(2,36,41,8,24,47,14,30,53),(3,42,25,15,54,19,9,48,31),(4,26,37,10,32,43,16,20,49),(5,38,33,17,50,27,11,44,21),(6,34,51,12,22,39,18,28,45)], [(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)]])

90 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Q 6A ··· 6H 9A ··· 9R 9S ··· 9AS 18A ··· 18R order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 9 1 ··· 1 2 ··· 2 9 ··· 9 3 ··· 3 6 ··· 6 9 ··· 9

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 6 6 6 type + + + + image C1 C2 C3 C3 C6 C6 C9 C18 S3 C3×S3 C3×S3 S3×C9 C9⋊C6 C9⋊C18 C3×C9⋊C6 kernel C3×C9⋊C18 C3×C9⋊C9 C9⋊C18 C32×D9 C9⋊C9 C32×C9 C3×D9 C3×C9 C32×C9 C3×C9 C33 C32 C32 C3 C3 # reps 1 1 6 2 6 2 18 18 1 6 2 18 1 6 2

Matrix representation of C3×C9⋊C18 in GL8(𝔽19)

 11 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 11 0 0 0 0 0 0 0 11 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 7 0 0 0 0 0 1 0 0 0 0 0 0 0 7 6 18 18 18 6 0 0 0 0 0 11 0 0 0 0 14 15 15 8 8 1
,
 9 14 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 11 4 12 12 12 4 0 0 0 0 0 7 0 0 0 0 0 1 0 0 0 0 0 0 0 0 7 0 0 0 0 0 6 12 5 5 9 15

G:=sub<GL(8,GF(19))| [11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[11,11,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,1,7,0,14,0,0,7,0,0,6,0,15,0,0,0,7,0,18,0,15,0,0,0,0,0,18,11,8,0,0,0,0,0,18,0,8,0,0,0,0,0,6,0,1],[9,0,0,0,0,0,0,0,14,10,0,0,0,0,0,0,0,0,0,11,0,0,0,6,0,0,0,4,0,1,0,12,0,0,0,12,0,0,7,5,0,0,0,12,7,0,0,5,0,0,1,12,0,0,0,9,0,0,0,4,0,0,0,15] >;

C3×C9⋊C18 in GAP, Magma, Sage, TeX

C_3\times C_9\rtimes C_{18}
% in TeX

G:=Group("C3xC9:C18");
// GroupNames label

G:=SmallGroup(486,96);
// by ID

G=gap.SmallGroup(486,96);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,115,8104,3250,208,11669]);
// Polycyclic

G:=Group<a,b,c|a^3=b^9=c^18=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^2>;
// generators/relations

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