Copied to
clipboard

## G = C9×Dic9order 324 = 22·34

### Direct product of C9 and Dic9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C9×Dic9
 Chief series C1 — C3 — C9 — C3×C9 — C3×C18 — C9×C18 — C9×Dic9
 Lower central C9 — C9×Dic9
 Upper central C1 — C18

Generators and relations for C9×Dic9
G = < a,b,c | a9=b18=1, c2=b9, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C9×Dic9
On 36 points
Generators in S36
(1 3 5 7 9 11 13 15 17)(2 4 6 8 10 12 14 16 18)(19 35 33 31 29 27 25 23 21)(20 36 34 32 30 28 26 24 22)
(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 36 10 27)(2 35 11 26)(3 34 12 25)(4 33 13 24)(5 32 14 23)(6 31 15 22)(7 30 16 21)(8 29 17 20)(9 28 18 19)

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

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

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

108 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 9A ··· 9F 9G ··· 9AM 12A 12B 12C 12D 18A ··· 18F 18G ··· 18AM 36A ··· 36L order 1 2 3 3 3 3 3 4 4 6 6 6 6 6 9 ··· 9 9 ··· 9 12 12 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 2 9 9 1 1 2 2 2 1 ··· 1 2 ··· 2 9 9 9 9 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 C3 C4 C6 C9 C12 C18 C36 S3 Dic3 D9 C3×S3 Dic9 C3×Dic3 C3×D9 S3×C9 C3×Dic9 C9×Dic3 C9×D9 C9×Dic9 kernel C9×Dic9 C9×C18 C3×Dic9 C92 C3×C18 Dic9 C3×C9 C18 C9 C3×C18 C3×C9 C18 C3×C6 C9 C32 C6 C6 C3 C3 C2 C1 # reps 1 1 2 2 2 6 4 6 12 1 1 3 2 3 2 6 6 6 6 18 18

Matrix representation of C9×Dic9 in GL2(𝔽19) generated by

 4 0 0 4
,
 14 0 0 15
,
 0 18 1 0
G:=sub<GL(2,GF(19))| [4,0,0,4],[14,0,0,15],[0,1,18,0] >;

C9×Dic9 in GAP, Magma, Sage, TeX

C_9\times {\rm Dic}_9
% in TeX

G:=Group("C9xDic9");
// GroupNames label

G:=SmallGroup(324,6);
// by ID

G=gap.SmallGroup(324,6);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,79,5404,208,7781]);
// Polycyclic

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

Export

׿
×
𝔽