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G = D9×C18  order 324 = 22·34

Direct product of C18 and D9

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

Aliases: D9×C18, C18⋊3C18, C92⋊2C22, C9⋊3(C2×C18), (C9×C18)⋊1C2, C6.4(S3×C9), (C3×C9).6D6, C6.9(C3×D9), C3.4(C6×D9), C3.1(S3×C18), (C6×D9).2C3, (C3×D9).2C6, (C3×C18).16S3, (C3×C18).25C6, C32.13(S3×C6), (C3×C9).6(C2×C6), (C3×C6).29(C3×S3), SmallGroup(324,61)

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, C3, C3, C22, S3, C6, C6, C9, C9, C32, D6, C2×C6, D9, C18, C18, C3×S3, C3×C6, C3×C9, C3×C9, D18, C2×C18, S3×C6, C3×D9, S3×C9, C3×C18, C3×C18, C92, C6×D9, S3×C18, C9×D9, C9×C18, D9×C18
Quotients: C1, C2, C3, C22, S3, C6, C9, D6, C2×C6, D9, C18, 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 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+++++++
imageC1C2C2C3C6C6C9C18C18S3D6D9C3×S3D18S3×C6C3×D9S3×C9C6×D9S3×C18C9×D9D9×C18
kernelD9×C18C9×D9C9×C18C6×D9C3×D9C3×C18D18D9C18C3×C18C3×C9C18C3×C6C9C32C6C6C3C3C2C1
# reps121242612611323266661818

Matrix representation of D9×C18 ►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] >;
 

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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