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## G = (C3×C9)⋊C18order 486 = 2·35

### 2nd semidirect product of C3×C9 and C18 acting via C18/C3=C6

Aliases: C9⋊S32C9, (C3×C9)⋊2C18, C32⋊C9.5S3, (C32×C9).4C6, C32.7(S3×C9), C33.50(C3×S3), C3.3(C32⋊C18), C32.19He31C2, C32.36(C32⋊C6), C3.2(He3.S3), (C3×C9⋊S3).2C3, SmallGroup(486,20)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — (C3×C9)⋊C18
 Chief series C1 — C3 — C32 — C3×C9 — C32×C9 — C32.19He3 — (C3×C9)⋊C18
 Lower central C3×C9 — (C3×C9)⋊C18
 Upper central C1 — C3

Generators and relations for (C3×C9)⋊C18
G = < a,b,c | a3=b9=c18=1, ab=ba, cac-1=a-1b3, cbc-1=ab2 >

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

39 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 6A 6B 9A ··· 9I 9J ··· 9O 9P ··· 9U 18A ··· 18F order 1 2 3 3 3 3 3 3 3 3 6 6 9 ··· 9 9 ··· 9 9 ··· 9 18 ··· 18 size 1 27 1 1 2 2 2 6 6 6 27 27 6 ··· 6 9 ··· 9 18 ··· 18 27 ··· 27

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 6 6 6 6 type + + + + + image C1 C2 C3 C6 C9 C18 S3 C3×S3 S3×C9 C32⋊C6 C32⋊C18 He3.S3 (C3×C9)⋊C18 kernel (C3×C9)⋊C18 C32.19He3 C3×C9⋊S3 C32×C9 C9⋊S3 C3×C9 C32⋊C9 C33 C32 C32 C3 C3 C1 # reps 1 1 2 2 6 6 1 2 6 1 2 3 6

Matrix representation of (C3×C9)⋊C18 in GL6(𝔽19)

 1 0 0 0 0 0 0 7 0 0 0 0 18 11 11 0 0 0 0 0 0 1 0 0 12 8 0 7 11 0 8 0 0 18 0 7
,
 16 0 0 0 0 0 0 17 0 0 0 0 17 0 17 0 0 0 0 0 0 6 0 0 3 3 0 4 9 0 16 16 0 9 0 9
,
 11 12 0 1 0 13 0 0 0 11 0 0 1 8 0 7 11 7 12 12 13 0 0 0 8 0 8 11 0 11 0 8 18 8 0 8

G:=sub<GL(6,GF(19))| [1,0,18,0,12,8,0,7,11,0,8,0,0,0,11,0,0,0,0,0,0,1,7,18,0,0,0,0,11,0,0,0,0,0,0,7],[16,0,17,0,3,16,0,17,0,0,3,16,0,0,17,0,0,0,0,0,0,6,4,9,0,0,0,0,9,0,0,0,0,0,0,9],[11,0,1,12,8,0,12,0,8,12,0,8,0,0,0,13,8,18,1,11,7,0,11,8,0,0,11,0,0,0,13,0,7,0,11,8] >;

(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(486,20);
// by ID

G=gap.SmallGroup(486,20);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,8643,873,237,3244,3250,11669]);
// Polycyclic

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

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