Copied to
clipboard

G = D9xC18  order 324 = 22·34

Direct product of C18 and D9

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

Aliases: D9xC18, C18:3C18, C92:2C22, C9:3(C2xC18), (C9xC18):1C2, C6.4(S3xC9), (C3xC9).6D6, C6.9(C3xD9), C3.4(C6xD9), C3.1(S3xC18), (C6xD9).2C3, (C3xD9).2C6, (C3xC18).16S3, (C3xC18).25C6, C32.13(S3xC6), (C3xC9).6(C2xC6), (C3xC6).29(C3xS3), SmallGroup(324,61)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C9 — D9xC18
Chief series
C1 — C3 — C9 — C3xC9 — C92 — C9xD9 — D9xC18
Lower central
C9 — D9xC18
Upper central
C1 — C18

Generators and relations for D9xC18
 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, C3, C3, C22, S3, C6, C6, C9, C9, C32, D6, C2xC6, D9, C18, C18, C3xS3, C3xC6, C3xC9, C3xC9, D18, C2xC18, S3xC6, C3xD9, S3xC9, C3xC18, C3xC18, C92, C6xD9, S3xC18, C9xD9, C9xC18, D9xC18
Quotients: C1, C2, C3, C22, S3, C6, C9, D6, C2xC6, D9, C18, C3xS3, D18, C2xC18, S3xC6, C3xD9, S3xC9, C6xD9, S3xC18, C9xD9, D9xC18

Smallest permutation representation of D9xC18
►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 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 19)(9 20)(10 21)(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)
 
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), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,19)(9,20)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)>;
 
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), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,19)(9,20)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29) );
 
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)], [(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,19),(9,20),(10,21),(11,22),(12,23),(13,24),(14,25),(15,26),(16,27),(17,28),(18,29)]])
 

108 conjugacy classes

class 1 2A2B2C3A3B3C3D3E6A6B6C6D6E6F6G6H6I9A···9F9G···9AM18A···18F18G···18AM18AN···18AY
order1222333336666666669···99···918···1818···1818···18
size1199112221122299991···12···21···12···29···9

108 irreducible representations

dim111111111222222222222
type+++++++
imageC1C2C2C3C6C6C9C18C18S3D6D9C3xS3D18S3xC6C3xD9S3xC9C6xD9S3xC18C9xD9D9xC18
kernelD9xC18C9xD9C9xC18C6xD9C3xD9C3xC18D18D9C18C3xC18C3xC9C18C3xC6C9C32C6C6C3C3C2C1
# reps121242612611323266661818

Matrix representation of D9xC18 ►in GL2(F19) generated by

130
013
,
90
017
,
016
60
G:=sub<GL(2,GF(19))| [13,0,0,13],[9,0,0,17],[0,6,16,0] >;
 

D9xC18 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
 

׿
x
:
Z
F
o
wr
Q
<