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G = C3×C32⋊C18  order 486 = 2·35

Direct product of C3 and C32⋊C18

direct product, metabelian, supersoluble, monomial

Aliases: C3×C32⋊C18, C33⋊2C18, C34.2C6, (C32×C9)⋊2S3, C32⋊C9⋊17C6, C32⋊2(C3×C18), C32.16(S3×C9), C33.20(C3×C6), C33.75(C3×S3), C32.28(S3×C32), C32.48(C32⋊C6), (C3×C3⋊S3)⋊C9, C3.2(S3×C3×C9), C3⋊S3⋊2(C3×C9), (C3×C9)⋊9(C3×S3), (C3×C32⋊C9)⋊1C2, (C32×C3⋊S3).1C3, (C3×C3⋊S3).1C32, C3.10(C3×C32⋊C6), SmallGroup(486,93)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3×C32⋊C18
Chief series
C1 — C3 — C32 — C33 — C34 — C3×C32⋊C9 — C3×C32⋊C18
Lower central
C32 — C3×C32⋊C18
Upper central
C1 — C32

Generators and relations for C3×C32⋊C18
 G = < a,b,c,d | a3=b3=c3=d18=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c-1, dcd-1=c-1 >

Subgroups: 472 in 126 conjugacy classes, 36 normal (15 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C32, C32, C32, C18, C3×S3, C3⋊S3, C3×C6, C3×C9, C3×C9, C33, C33, C33, S3×C9, C3×C18, S3×C32, C3×C3⋊S3, C3×C3⋊S3, C32⋊C9, C32⋊C9, C32×C9, C32×C9, C34, C32⋊C18, S3×C3×C9, C32×C3⋊S3, C3×C32⋊C9, C3×C32⋊C18
Quotients: C1, C2, C3, S3, C6, C9, C32, C18, C3×S3, C3×C6, C3×C9, S3×C9, C32⋊C6, C3×C18, S3×C32, C32⋊C18, S3×C3×C9, C3×C32⋊C6, C3×C32⋊C18

Smallest permutation representation of C3×C32⋊C18
►On 54 points
Generators in S54
(1 28 43)(2 29 44)(3 30 45)(4 31 46)(5 32 47)(6 33 48)(7 34 49)(8 35 50)(9 36 51)(10 19 52)(11 20 53)(12 21 54)(13 22 37)(14 23 38)(15 24 39)(16 25 40)(17 26 41)(18 27 42)

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

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

(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)| (1,28,43)(2,29,44)(3,30,45)(4,31,46)(5,32,47)(6,33,48)(7,34,49)(8,35,50)(9,36,51)(10,19,52)(11,20,53)(12,21,54)(13,22,37)(14,23,38)(15,24,39)(16,25,40)(17,26,41)(18,27,42), (1,28,43)(2,8,14)(3,51,24)(4,46,31)(5,17,11)(6,27,54)(7,34,49)(9,39,30)(10,52,19)(12,33,42)(13,22,37)(15,45,36)(16,40,25)(18,21,48)(20,32,26)(23,29,35)(38,44,50)(41,53,47), (1,37,34)(2,35,38)(3,39,36)(4,19,40)(5,41,20)(6,21,42)(7,43,22)(8,23,44)(9,45,24)(10,25,46)(11,47,26)(12,27,48)(13,49,28)(14,29,50)(15,51,30)(16,31,52)(17,53,32)(18,33,54), (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( (1,28,43)(2,29,44)(3,30,45)(4,31,46)(5,32,47)(6,33,48)(7,34,49)(8,35,50)(9,36,51)(10,19,52)(11,20,53)(12,21,54)(13,22,37)(14,23,38)(15,24,39)(16,25,40)(17,26,41)(18,27,42), (1,28,43)(2,8,14)(3,51,24)(4,46,31)(5,17,11)(6,27,54)(7,34,49)(9,39,30)(10,52,19)(12,33,42)(13,22,37)(15,45,36)(16,40,25)(18,21,48)(20,32,26)(23,29,35)(38,44,50)(41,53,47), (1,37,34)(2,35,38)(3,39,36)(4,19,40)(5,41,20)(6,21,42)(7,43,22)(8,23,44)(9,45,24)(10,25,46)(11,47,26)(12,27,48)(13,49,28)(14,29,50)(15,51,30)(16,31,52)(17,53,32)(18,33,54), (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([[(1,28,43),(2,29,44),(3,30,45),(4,31,46),(5,32,47),(6,33,48),(7,34,49),(8,35,50),(9,36,51),(10,19,52),(11,20,53),(12,21,54),(13,22,37),(14,23,38),(15,24,39),(16,25,40),(17,26,41),(18,27,42)], [(1,28,43),(2,8,14),(3,51,24),(4,46,31),(5,17,11),(6,27,54),(7,34,49),(9,39,30),(10,52,19),(12,33,42),(13,22,37),(15,45,36),(16,40,25),(18,21,48),(20,32,26),(23,29,35),(38,44,50),(41,53,47)], [(1,37,34),(2,35,38),(3,39,36),(4,19,40),(5,41,20),(6,21,42),(7,43,22),(8,23,44),(9,45,24),(10,25,46),(11,47,26),(12,27,48),(13,49,28),(14,29,50),(15,51,30),(16,31,52),(17,53,32),(18,33,54)], [(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···3Q3R···3Z6A···6H9A···9R9S···9AJ18A···18R
order123···33···33···36···69···99···918···18
size191···12···26···69···93···36···69···9

90 irreducible representations

dim111111112222666
type++++
imageC1C2C3C3C6C6C9C18S3C3×S3C3×S3S3×C9C32⋊C6C32⋊C18C3×C32⋊C6
kernelC3×C32⋊C18C3×C32⋊C9C32⋊C18C32×C3⋊S3C32⋊C9C34C3×C3⋊S3C33C32×C9C3×C9C33C32C32C3C3
# reps116262181816218162

Matrix representation of C3×C32⋊C18 ►in GL8(𝔽19)

110000000
011000000
001100000
000110000
000011000
000001100
000000110
000000011
,
70000000
1211000000
001100000
00070000
00811000
00000700
000000110
00182016131
,
10000000
01000000
00700000
00070000
00007000
000001100
000000110
008500011
,
1317000000
106000000
00000070
008502104
0014401167
00070000
0012124000
0015397214

G:=sub<GL(8,GF(19))| [11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11],[7,12,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,8,0,0,18,0,0,0,7,1,0,0,2,0,0,0,0,1,0,0,0,0,0,0,0,0,7,0,16,0,0,0,0,0,0,11,13,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,0,0,0,0,8,0,0,0,7,0,0,0,5,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11],[13,10,0,0,0,0,0,0,17,6,0,0,0,0,0,0,0,0,0,8,14,0,12,15,0,0,0,5,4,7,12,3,0,0,0,0,0,0,4,9,0,0,0,2,11,0,0,7,0,0,7,10,6,0,0,2,0,0,0,4,7,0,0,14] >;
 

C3×C32⋊C18 in GAP, Magma, Sage, TeX 

C_3\times C_3^2\rtimes C_{18}
 
% in TeX
 
G:=Group("C3xC3^2:C18");
 
// GroupNames label
 
G:=SmallGroup(486,93);
 
// by ID
 
G=gap.SmallGroup(486,93);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,115,3244,3250,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^3=d^18=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c^-1,d*c*d^-1=c^-1>;
 
// generators/relations
 

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