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## G = C32×C9⋊C6order 486 = 2·35

### Direct product of C32 and C9⋊C6

Aliases: C32×C9⋊C6, D9⋊C33, C34.9S3, C9⋊(C32×C6), (C3×D9)⋊C32, (C32×C9)⋊18C6, (C32×D9)⋊3C3, C3.3(S3×C33), C33.46(C3×S3), C32.9(S3×C32), 3- 1+25(C3×C6), (C3×3- 1+2)⋊19C6, (C32×3- 1+2)⋊1C2, (C3×C9)⋊12(C3×C6), SmallGroup(486,224)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C32×C9⋊C6
 Chief series C1 — C3 — C9 — C3×C9 — C32×C9 — C32×3- 1+2 — C32×C9⋊C6
 Lower central C9 — C32×C9⋊C6
 Upper central C1 — C32

Generators and relations for C32×C9⋊C6
G = < a,b,c,d | a3=b3=c9=d6=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c2 >

Subgroups: 760 in 270 conjugacy classes, 90 normal (11 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C3×S3, C3×C6, C3×C9, C3×C9, 3- 1+2, 3- 1+2, C33, C33, C33, C3×D9, C9⋊C6, S3×C32, C32×C6, C32×C9, C32×C9, C3×3- 1+2, C3×3- 1+2, C34, C32×D9, C3×C9⋊C6, S3×C33, C32×3- 1+2, C32×C9⋊C6
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, C33, C9⋊C6, S3×C32, C32×C6, C3×C9⋊C6, S3×C33, C32×C9⋊C6

Smallest permutation representation of C32×C9⋊C6
On 54 points
Generators in S54
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)
(1 20 11)(2 21 12)(3 22 13)(4 23 14)(5 24 15)(6 25 16)(7 26 17)(8 27 18)(9 19 10)(28 46 37)(29 47 38)(30 48 39)(31 49 40)(32 50 41)(33 51 42)(34 52 43)(35 53 44)(36 54 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)
(1 37 23 34 17 49)(2 42 21 33 12 51)(3 38 19 32 16 53)(4 43 26 31 11 46)(5 39 24 30 15 48)(6 44 22 29 10 50)(7 40 20 28 14 52)(8 45 27 36 18 54)(9 41 25 35 13 47)

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

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

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

90 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Q 3R ··· 3AI 6A ··· 6Z 9A ··· 9AA order 1 2 3 ··· 3 3 ··· 3 3 ··· 3 6 ··· 6 9 ··· 9 size 1 9 1 ··· 1 2 ··· 2 3 ··· 3 9 ··· 9 6 ··· 6

90 irreducible representations

 dim 1 1 1 1 1 1 2 2 6 6 type + + + + image C1 C2 C3 C3 C6 C6 S3 C3×S3 C9⋊C6 C3×C9⋊C6 kernel C32×C9⋊C6 C32×3- 1+2 C32×D9 C3×C9⋊C6 C32×C9 C3×3- 1+2 C34 C33 C32 C3 # reps 1 1 2 24 2 24 1 26 1 8

Matrix representation of C32×C9⋊C6 in GL8(𝔽19)

 11 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 7
,
 7 0 0 0 0 0 0 0 0 7 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
,
 18 1 0 0 0 0 0 0 18 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 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
,
 11 8 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 1 0 0 11 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 1 0 0 0

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

C32×C9⋊C6 in GAP, Magma, Sage, TeX

C_3^2\times C_9\rtimes C_6
% in TeX

G:=Group("C3^2xC9:C6");
// GroupNames label

G:=SmallGroup(486,224);
// by ID

G=gap.SmallGroup(486,224);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,8104,3250,208,11669]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^9=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^2>;
// generators/relations

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