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G = C3×C9⋊C18  order 486 = 2·35

Direct product of C3 and C9⋊C18

direct product, metabelian, supersoluble, monomial

Aliases: C3×C9⋊C18, (C3×D9)⋊C9, C9⋊C9⋊13C6, (C3×C9)⋊4C18, C9⋊2(C3×C18), D9⋊2(C3×C9), (C32×C9).7C6, (C32×C9).4S3, C33.76(C3×S3), C32.17(S3×C9), (C3×D9).1C32, (C32×D9).2C3, C32.12(C9⋊C6), C32.30(S3×C32), (C3×C9⋊C9)⋊1C2, C3.3(S3×C3×C9), C3.3(C3×C9⋊C6), (C3×C9).10(C3×C6), (C3×C9).14(C3×S3), SmallGroup(486,96)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C9 — C3×C9⋊C18
Chief series
C1 — C3 — C9 — C3×C9 — C32×C9 — C3×C9⋊C9 — C3×C9⋊C18
Lower central
C9 — C3×C9⋊C18
Upper central
C1 — C32

Generators and relations for C3×C9⋊C18
 G = < a,b,c | a3=b9=c18=1, ab=ba, ac=ca, cbc-1=b2 >

Subgroups: 256 in 90 conjugacy classes, 36 normal (15 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C18, C3×S3, C3×C6, C3×C9, C3×C9, C3×C9, C33, C3×D9, C3×D9, S3×C9, C3×C18, S3×C32, C9⋊C9, C9⋊C9, C32×C9, C32×C9, C9⋊C18, C32×D9, S3×C3×C9, C3×C9⋊C9, C3×C9⋊C18
Quotients: C1, C2, C3, S3, C6, C9, C32, C18, C3×S3, C3×C6, C3×C9, S3×C9, C9⋊C6, C3×C18, S3×C32, C9⋊C18, S3×C3×C9, C3×C9⋊C6, C3×C9⋊C18

Smallest permutation representation of C3×C9⋊C18
►On 54 points
Generators in S54
(1 21 42)(2 22 43)(3 23 44)(4 24 45)(5 25 46)(6 26 47)(7 27 48)(8 28 49)(9 29 50)(10 30 51)(11 31 52)(12 32 53)(13 33 54)(14 34 37)(15 35 38)(16 36 39)(17 19 40)(18 20 41)

(1 46 35 13 40 29 7 52 23)(2 36 41 8 24 47 14 30 53)(3 42 25 15 54 19 9 48 31)(4 26 37 10 32 43 16 20 49)(5 38 33 17 50 27 11 44 21)(6 34 51 12 22 39 18 28 45)

(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)(37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54)
 
G:=sub<Sym(54)| (1,21,42)(2,22,43)(3,23,44)(4,24,45)(5,25,46)(6,26,47)(7,27,48)(8,28,49)(9,29,50)(10,30,51)(11,31,52)(12,32,53)(13,33,54)(14,34,37)(15,35,38)(16,36,39)(17,19,40)(18,20,41), (1,46,35,13,40,29,7,52,23)(2,36,41,8,24,47,14,30,53)(3,42,25,15,54,19,9,48,31)(4,26,37,10,32,43,16,20,49)(5,38,33,17,50,27,11,44,21)(6,34,51,12,22,39,18,28,45), (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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)>;
 
G:=Group( (1,21,42)(2,22,43)(3,23,44)(4,24,45)(5,25,46)(6,26,47)(7,27,48)(8,28,49)(9,29,50)(10,30,51)(11,31,52)(12,32,53)(13,33,54)(14,34,37)(15,35,38)(16,36,39)(17,19,40)(18,20,41), (1,46,35,13,40,29,7,52,23)(2,36,41,8,24,47,14,30,53)(3,42,25,15,54,19,9,48,31)(4,26,37,10,32,43,16,20,49)(5,38,33,17,50,27,11,44,21)(6,34,51,12,22,39,18,28,45), (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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54) );
 
G=PermutationGroup([[(1,21,42),(2,22,43),(3,23,44),(4,24,45),(5,25,46),(6,26,47),(7,27,48),(8,28,49),(9,29,50),(10,30,51),(11,31,52),(12,32,53),(13,33,54),(14,34,37),(15,35,38),(16,36,39),(17,19,40),(18,20,41)], [(1,46,35,13,40,29,7,52,23),(2,36,41,8,24,47,14,30,53),(3,42,25,15,54,19,9,48,31),(4,26,37,10,32,43,16,20,49),(5,38,33,17,50,27,11,44,21),(6,34,51,12,22,39,18,28,45)], [(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),(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)]])
 

90 conjugacy classes

class 1  2 3A···3H3I···3Q6A···6H9A···9R9S···9AS18A···18R
order123···33···36···69···99···918···18
size191···12···29···93···36···69···9

90 irreducible representations

dim111111112222666
type++++
imageC1C2C3C3C6C6C9C18S3C3×S3C3×S3S3×C9C9⋊C6C9⋊C18C3×C9⋊C6
kernelC3×C9⋊C18C3×C9⋊C9C9⋊C18C32×D9C9⋊C9C32×C9C3×D9C3×C9C32×C9C3×C9C33C32C32C3C3
# reps116262181816218162

Matrix representation of C3×C9⋊C18 ►in GL8(𝔽19)

110000000
011000000
00100000
00010000
00001000
00000100
00000010
00000001
,
110000000
117000000
00070000
00007000
00100000
00761818186
000001100
00141515881
,
914000000
010000000
00000010
001141212124
00000700
00010000
00007000
0061255915

G:=sub<GL(8,GF(19))| [11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[11,11,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,1,7,0,14,0,0,7,0,0,6,0,15,0,0,0,7,0,18,0,15,0,0,0,0,0,18,11,8,0,0,0,0,0,18,0,8,0,0,0,0,0,6,0,1],[9,0,0,0,0,0,0,0,14,10,0,0,0,0,0,0,0,0,0,11,0,0,0,6,0,0,0,4,0,1,0,12,0,0,0,12,0,0,7,5,0,0,0,12,7,0,0,5,0,0,1,12,0,0,0,9,0,0,0,4,0,0,0,15] >;
 

C3×C9⋊C18 in GAP, Magma, Sage, TeX 

C_3\times C_9\rtimes C_{18}
 
% in TeX
 
G:=Group("C3xC9:C18");
 
// GroupNames label
 
G:=SmallGroup(486,96);
 
// by ID
 
G=gap.SmallGroup(486,96);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,115,8104,3250,208,11669]);
 
// Polycyclic
 
G:=Group<a,b,c|a^3=b^9=c^18=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^2>;
 
// generators/relations
 

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