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G = C2×C9⋊C18  order 324 = 22·34

Direct product of C2 and C9⋊C18

direct product, metacyclic, supersoluble, monomial

Aliases: C2×C9⋊C18, C18⋊C18, D9⋊C18, D18⋊C9, C9⋊(C2×C18), C9⋊C9⋊C22, (C3×D9).C6, (C6×D9).C3, C6.6(S3×C9), (C3×C9).1D6, C6.5(C9⋊C6), C3.3(S3×C18), (C3×C18).5S3, (C3×C18).5C6, C32.16(S3×C6), (C2×C9⋊C9)⋊C2, (C3×C9).(C2×C6), C3.2(C2×C9⋊C6), (C3×C6).32(C3×S3), SmallGroup(324,64)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C9 — C2×C9⋊C18
Chief series
C1 — C3 — C9 — C3×C9 — C9⋊C9 — C9⋊C18 — C2×C9⋊C18
Lower central
C9 — C2×C9⋊C18
Upper central
C1 — C6

Generators and relations for C2×C9⋊C18
 G = < a,b,c | a2=b9=c18=1, ab=ba, ac=ca, cbc-1=b2 >

C1
C2
9C2
9C2
C3
C3
2C3
9C22
C6
C6
2C6
3S3
3S3
9C6
9C6
C9
C32
2C9
3C9
6C9
3D6
9C2×C6
C3×C6
C18
D9
D9
2C18
3C18
3C3×S3
3C3×S3
6C18
9C18
9C18
C3×C9
C3×C9
2C3×C9
D18
3S3×C6
9C2×C18
C3×C18
C3×D9
C3×D9
C3×C18
2C3×C18
3S3×C9
3S3×C9
C9⋊C9
C6×D9
3S3×C18
C9⋊C18
C9⋊C18
C2×C9⋊C9
C2×C9⋊C18

Smallest permutation representation of C2×C9⋊C18
►On 36 points
Generators in S36
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)

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

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

60 conjugacy classes

class 1 2A2B2C3A3B3C3D3E6A6B6C6D6E6F6G6H6I9A···9F9G···9O18A···18F18G···18O18P···18AA
order1222333336666666669···99···918···1818···1818···18
size1199112221122299993···36···63···36···69···9

60 irreducible representations

dim1111111112222226666
type+++++++
imageC1C2C2C3C6C6C9C18C18S3D6C3×S3S3×C6S3×C9S3×C18C9⋊C6C2×C9⋊C6C9⋊C18C2×C9⋊C18
kernelC2×C9⋊C18C9⋊C18C2×C9⋊C9C6×D9C3×D9C3×C18D18D9C18C3×C18C3×C9C3×C6C32C6C3C6C3C2C1
# reps12124261261122661122

Matrix representation of C2×C9⋊C18 ►in GL8(𝔽19)

180000000
018000000
00100000
00010000
00001000
00000100
00000010
00000001
,
110000000
117000000
00070000
00007000
00100000
00000001
000001100
000000110
,
84000000
011000000
00000070
000000011
000001100
00070000
000011000
001100000

G:=sub<GL(8,GF(19))| [18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[11,11,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,1,0,0],[8,0,0,0,0,0,0,0,4,11,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0] >;
 

C2×C9⋊C18 in GAP, Magma, Sage, TeX 

C_2\times C_9\rtimes C_{18}
 
% in TeX
 
G:=Group("C2xC9:C18");
 
// GroupNames label
 
G:=SmallGroup(324,64);
 
// by ID
 
G=gap.SmallGroup(324,64);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,68,5404,1096,208,7781]);
 
// Polycyclic
 
G:=Group<a,b,c|a^2=b^9=c^18=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^2>;
 
// generators/relations
 

Export

Subgroup lattice of C2×C9⋊C18 in TeX

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