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## G = D9×C18order 324 = 22·34

### Direct product of C18 and D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — D9×C18
 Chief series C1 — C3 — C9 — C3×C9 — C92 — C9×D9 — D9×C18
 Lower central C9 — D9×C18
 Upper central C1 — C18

Generators and relations for D9×C18
G = < a,b,c | a18=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 163 in 59 conjugacy classes, 27 normal (21 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C22, S3 [×2], C6 [×2], C6 [×3], C9 [×2], C9 [×5], C32, D6, C2×C6, D9 [×2], C18 [×2], C18 [×7], C3×S3 [×2], C3×C6, C3×C9 [×2], C3×C9, D18, C2×C18, S3×C6, C3×D9 [×2], S3×C9 [×2], C3×C18 [×2], C3×C18, C92, C6×D9, S3×C18, C9×D9 [×2], C9×C18, D9×C18
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], C9, D6, C2×C6, D9, C18 [×3], C3×S3, D18, C2×C18, S3×C6, C3×D9, S3×C9, C6×D9, S3×C18, C9×D9, D9×C18

Smallest permutation representation of D9×C18
On 36 points
Generators in S36
(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)
(1 5 9 13 17 3 7 11 15)(2 6 10 14 18 4 8 12 16)(19 33 29 25 21 35 31 27 23)(20 34 30 26 22 36 32 28 24)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)

G:=sub<Sym(36)| (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), (1,5,9,13,17,3,7,11,15)(2,6,10,14,18,4,8,12,16)(19,33,29,25,21,35,31,27,23)(20,34,30,26,22,36,32,28,24), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)>;

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), (1,5,9,13,17,3,7,11,15)(2,6,10,14,18,4,8,12,16)(19,33,29,25,21,35,31,27,23)(20,34,30,26,22,36,32,28,24), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36) );

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)], [(1,5,9,13,17,3,7,11,15),(2,6,10,14,18,4,8,12,16),(19,33,29,25,21,35,31,27,23),(20,34,30,26,22,36,32,28,24)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36)])

108 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 6A 6B 6C 6D 6E 6F 6G 6H 6I 9A ··· 9F 9G ··· 9AM 18A ··· 18F 18G ··· 18AM 18AN ··· 18AY order 1 2 2 2 3 3 3 3 3 6 6 6 6 6 6 6 6 6 9 ··· 9 9 ··· 9 18 ··· 18 18 ··· 18 18 ··· 18 size 1 1 9 9 1 1 2 2 2 1 1 2 2 2 9 9 9 9 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 9 ··· 9

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C3 C6 C6 C9 C18 C18 S3 D6 D9 C3×S3 D18 S3×C6 C3×D9 S3×C9 C6×D9 S3×C18 C9×D9 D9×C18 kernel D9×C18 C9×D9 C9×C18 C6×D9 C3×D9 C3×C18 D18 D9 C18 C3×C18 C3×C9 C18 C3×C6 C9 C32 C6 C6 C3 C3 C2 C1 # reps 1 2 1 2 4 2 6 12 6 1 1 3 2 3 2 6 6 6 6 18 18

Matrix representation of D9×C18 in GL2(𝔽19) generated by

 13 0 0 13
,
 9 0 0 17
,
 0 16 6 0
G:=sub<GL(2,GF(19))| [13,0,0,13],[9,0,0,17],[0,6,16,0] >;

D9×C18 in GAP, Magma, Sage, TeX

D_9\times C_{18}
% in TeX

G:=Group("D9xC18");
// GroupNames label

G:=SmallGroup(324,61);
// by ID

G=gap.SmallGroup(324,61);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,68,5404,208,7781]);
// Polycyclic

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

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