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G = C9×Dic9order 324 = 22·34

Direct product of C9 and Dic9

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C9×Dic9, C93C36, C922C4, C18.6D9, C18.5C18, C2.(C9×D9), C6.1(S3×C9), C6.8(C3×D9), (C3×C9).6C12, (C9×C18).1C2, (C3×C18).15S3, (C3×C18).24C6, C3.4(C3×Dic9), C3.1(C9×Dic3), (C3×C9).4Dic3, (C3×Dic9).2C3, C32.11(C3×Dic3), (C3×C6).25(C3×S3), SmallGroup(324,6)

Series: Derived Chief Lower central Upper central

C1C9 — C9×Dic9
C1C3C9C3×C9C3×C18C9×C18 — C9×Dic9
C9 — C9×Dic9
C1C18

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

2C3
9C4
2C6
2C9
2C9
2C9
2C9
2C9
3Dic3
9C12
2C18
2C18
2C18
2C18
2C18
2C3×C9
3C3×Dic3
9C36
2C3×C18
3C9×Dic3

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

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

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

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

108 conjugacy classes

class 1  2 3A3B3C3D3E4A4B6A6B6C6D6E9A···9F9G···9AM12A12B12C12D18A···18F18G···18AM36A···36L
order123333344666669···99···91212121218···1818···1836···36
size111122299112221···12···299991···12···29···9

108 irreducible representations

dim111111111222222222222
type+++-+-
imageC1C2C3C4C6C9C12C18C36S3Dic3D9C3×S3Dic9C3×Dic3C3×D9S3×C9C3×Dic9C9×Dic3C9×D9C9×Dic9
kernelC9×Dic9C9×C18C3×Dic9C92C3×C18Dic9C3×C9C18C9C3×C18C3×C9C18C3×C6C9C32C6C6C3C3C2C1
# reps112226461211323266661818

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

40
04
,
140
015
,
018
10
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

Subgroup lattice of C9×Dic9 in TeX

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