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## G = C6×D9order 108 = 22·33

### Direct product of C6 and D9

Aliases: C6×D9, C183C6, C32.3D6, C93(C2×C6), (C3×C18)⋊2C2, C3.1(S3×C6), C6.4(C3×S3), (C3×C6).7S3, (C3×C9)⋊3C22, SmallGroup(108,23)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C6×D9
 Chief series C1 — C3 — C9 — C3×C9 — C3×D9 — C6×D9
 Lower central C9 — C6×D9
 Upper central C1 — C6

Generators and relations for C6×D9
G = < a,b,c | a6=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C6×D9
On 36 points
Generators in S36
(1 11 4 14 7 17)(2 12 5 15 8 18)(3 13 6 16 9 10)(19 31 25 28 22 34)(20 32 26 29 23 35)(21 33 27 30 24 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 21)(2 20)(3 19)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 34)(11 33)(12 32)(13 31)(14 30)(15 29)(16 28)(17 36)(18 35)

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

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

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

C6×D9 is a maximal subgroup of   C3⋊D36  D6⋊D9

36 conjugacy classes

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

36 irreducible representations

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

Matrix representation of C6×D9 in GL2(𝔽19) generated by

 8 0 0 8
,
 9 0 4 17
,
 9 18 4 10
G:=sub<GL(2,GF(19))| [8,0,0,8],[9,4,0,17],[9,4,18,10] >;

C6×D9 in GAP, Magma, Sage, TeX

C_6\times D_9
% in TeX

G:=Group("C6xD9");
// GroupNames label

G:=SmallGroup(108,23);
// by ID

G=gap.SmallGroup(108,23);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-3,1203,138,1804]);
// Polycyclic

G:=Group<a,b,c|a^6=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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