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## G = C3×C33⋊S3order 486 = 2·35

### Direct product of C3 and C33⋊S3

Aliases: C3×C33⋊S3, C344S3, C3≀C36C6, (C3×He3)⋊9S3, He32(C3×S3), C338(C3×S3), C33.32(C3⋊S3), 3- 1+21(C3×S3), (C3×3- 1+2)⋊5S3, C32.4(He3⋊C2), (C3×C3≀C3)⋊1C2, C32.1(C3×C3⋊S3), C3.5(C3×He3⋊C2), SmallGroup(486,165)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3≀C3 — C3×C33⋊S3
 Chief series C1 — C3 — C32 — C33 — C3≀C3 — C3×C3≀C3 — C3×C33⋊S3
 Lower central C3≀C3 — C3×C33⋊S3
 Upper central C1 — C3

Generators and relations for C3×C33⋊S3
G = < a,b,c,d,e,f | a3=b3=c3=d3=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=bc=cb, bd=db, fbf=b-1, ece-1=cd=dc, fcf=cd-1, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 848 in 150 conjugacy classes, 20 normal (14 characteristic)
C1, C2, C3 [×2], C3 [×12], S3 [×5], C6 [×4], C9 [×4], C32 [×2], C32 [×2], C32 [×32], D9 [×2], C3×S3 [×17], C3⋊S3 [×2], C3×C6, C3×C9 [×2], He3, He3 [×3], 3- 1+2 [×2], 3- 1+2 [×6], C33 [×2], C33 [×11], C3×D9 [×2], C32⋊C6 [×3], C9⋊C6 [×6], S3×C32 [×4], C3×C3⋊S3 [×5], C3≀C3, C3≀C3 [×4], C3×He3, C3×3- 1+2 [×2], C34, C33⋊S3, C3×C32⋊C6, C3×C9⋊C6 [×2], C32×C3⋊S3, C3×C3≀C3, C3×C33⋊S3
Quotients: C1, C2, C3, S3 [×4], C6, C3×S3 [×4], C3⋊S3, He3⋊C2 [×3], C3×C3⋊S3, C33⋊S3, C3×He3⋊C2, C3×C33⋊S3

Permutation representations of C3×C33⋊S3
On 18 points - transitive group 18T169
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(7 9 8)(16 17 18)
(4 6 5)(7 8 9)(10 11 12)(16 18 17)
(1 3 2)(4 6 5)(7 9 8)(10 11 12)(13 14 15)(16 17 18)
(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)

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

G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (7,9,8)(16,17,18), (4,6,5)(7,8,9)(10,11,12)(16,18,17), (1,3,2)(4,6,5)(7,9,8)(10,11,12)(13,14,15)(16,17,18), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18) );

G=PermutationGroup([(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(7,9,8),(16,17,18)], [(4,6,5),(7,8,9),(10,11,12),(16,18,17)], [(1,3,2),(4,6,5),(7,9,8),(10,11,12),(13,14,15),(16,17,18)], [(1,4,7),(2,5,8),(3,6,9),(10,16,13),(11,17,14),(12,18,15)], [(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18)])

G:=TransitiveGroup(18,169);

On 27 points - transitive group 27T164
Generators in S27
(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)
(4 23 19)(5 24 20)(6 22 21)(10 15 18)(11 13 16)(12 14 17)
(1 8 26)(2 9 27)(3 7 25)(4 19 23)(5 20 24)(6 21 22)
(1 26 8)(2 27 9)(3 25 7)(4 19 23)(5 20 24)(6 21 22)(10 18 15)(11 16 13)(12 17 14)
(1 14 6)(2 15 4)(3 13 5)(7 16 24)(8 17 22)(9 18 23)(10 19 27)(11 20 25)(12 21 26)
(4 15)(5 13)(6 14)(7 25)(8 26)(9 27)(10 23)(11 24)(12 22)(16 20)(17 21)(18 19)

G:=sub<Sym(27)| (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), (4,23,19)(5,24,20)(6,22,21)(10,15,18)(11,13,16)(12,14,17), (1,8,26)(2,9,27)(3,7,25)(4,19,23)(5,20,24)(6,21,22), (1,26,8)(2,27,9)(3,25,7)(4,19,23)(5,20,24)(6,21,22)(10,18,15)(11,16,13)(12,17,14), (1,14,6)(2,15,4)(3,13,5)(7,16,24)(8,17,22)(9,18,23)(10,19,27)(11,20,25)(12,21,26), (4,15)(5,13)(6,14)(7,25)(8,26)(9,27)(10,23)(11,24)(12,22)(16,20)(17,21)(18,19)>;

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), (4,23,19)(5,24,20)(6,22,21)(10,15,18)(11,13,16)(12,14,17), (1,8,26)(2,9,27)(3,7,25)(4,19,23)(5,20,24)(6,21,22), (1,26,8)(2,27,9)(3,25,7)(4,19,23)(5,20,24)(6,21,22)(10,18,15)(11,16,13)(12,17,14), (1,14,6)(2,15,4)(3,13,5)(7,16,24)(8,17,22)(9,18,23)(10,19,27)(11,20,25)(12,21,26), (4,15)(5,13)(6,14)(7,25)(8,26)(9,27)(10,23)(11,24)(12,22)(16,20)(17,21)(18,19) );

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)], [(4,23,19),(5,24,20),(6,22,21),(10,15,18),(11,13,16),(12,14,17)], [(1,8,26),(2,9,27),(3,7,25),(4,19,23),(5,20,24),(6,21,22)], [(1,26,8),(2,27,9),(3,25,7),(4,19,23),(5,20,24),(6,21,22),(10,18,15),(11,16,13),(12,17,14)], [(1,14,6),(2,15,4),(3,13,5),(7,16,24),(8,17,22),(9,18,23),(10,19,27),(11,20,25),(12,21,26)], [(4,15),(5,13),(6,14),(7,25),(8,26),(9,27),(10,23),(11,24),(12,22),(16,20),(17,21),(18,19)])

G:=TransitiveGroup(27,164);

On 27 points - transitive group 27T178
Generators in S27
(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)
(1 11 13)(2 12 14)(3 10 15)(4 26 8)(5 27 9)(6 25 7)(16 22 19)(17 23 20)(18 24 21)
(1 10 14)(2 11 15)(3 12 13)(4 27 7)(5 25 8)(6 26 9)(16 18 17)(19 21 20)(22 24 23)
(1 13 11)(2 14 12)(3 15 10)(4 25 9)(5 26 7)(6 27 8)(16 23 21)(17 24 19)(18 22 20)
(1 20 6)(2 21 4)(3 19 5)(7 10 24)(8 11 22)(9 12 23)(13 18 27)(14 16 25)(15 17 26)
(4 21)(5 19)(6 20)(7 17)(8 18)(9 16)(10 15)(11 13)(12 14)(22 27)(23 25)(24 26)

G:=sub<Sym(27)| (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), (1,11,13)(2,12,14)(3,10,15)(4,26,8)(5,27,9)(6,25,7)(16,22,19)(17,23,20)(18,24,21), (1,10,14)(2,11,15)(3,12,13)(4,27,7)(5,25,8)(6,26,9)(16,18,17)(19,21,20)(22,24,23), (1,13,11)(2,14,12)(3,15,10)(4,25,9)(5,26,7)(6,27,8)(16,23,21)(17,24,19)(18,22,20), (1,20,6)(2,21,4)(3,19,5)(7,10,24)(8,11,22)(9,12,23)(13,18,27)(14,16,25)(15,17,26), (4,21)(5,19)(6,20)(7,17)(8,18)(9,16)(10,15)(11,13)(12,14)(22,27)(23,25)(24,26)>;

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), (1,11,13)(2,12,14)(3,10,15)(4,26,8)(5,27,9)(6,25,7)(16,22,19)(17,23,20)(18,24,21), (1,10,14)(2,11,15)(3,12,13)(4,27,7)(5,25,8)(6,26,9)(16,18,17)(19,21,20)(22,24,23), (1,13,11)(2,14,12)(3,15,10)(4,25,9)(5,26,7)(6,27,8)(16,23,21)(17,24,19)(18,22,20), (1,20,6)(2,21,4)(3,19,5)(7,10,24)(8,11,22)(9,12,23)(13,18,27)(14,16,25)(15,17,26), (4,21)(5,19)(6,20)(7,17)(8,18)(9,16)(10,15)(11,13)(12,14)(22,27)(23,25)(24,26) );

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)], [(1,11,13),(2,12,14),(3,10,15),(4,26,8),(5,27,9),(6,25,7),(16,22,19),(17,23,20),(18,24,21)], [(1,10,14),(2,11,15),(3,12,13),(4,27,7),(5,25,8),(6,26,9),(16,18,17),(19,21,20),(22,24,23)], [(1,13,11),(2,14,12),(3,15,10),(4,25,9),(5,26,7),(6,27,8),(16,23,21),(17,24,19),(18,22,20)], [(1,20,6),(2,21,4),(3,19,5),(7,10,24),(8,11,22),(9,12,23),(13,18,27),(14,16,25),(15,17,26)], [(4,21),(5,19),(6,20),(7,17),(8,18),(9,16),(10,15),(11,13),(12,14),(22,27),(23,25),(24,26)])

G:=TransitiveGroup(27,178);

39 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F ··· 3K 3L ··· 3T 3U 3V 3W 6A ··· 6H 9A ··· 9F order 1 2 3 3 3 3 3 3 ··· 3 3 ··· 3 3 3 3 6 ··· 6 9 ··· 9 size 1 27 1 1 2 2 2 3 ··· 3 6 ··· 6 18 18 18 27 ··· 27 18 ··· 18

39 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 3 6 6 type + + + + + + image C1 C2 C3 C6 S3 S3 S3 C3×S3 C3×S3 C3×S3 He3⋊C2 C33⋊S3 C3×C33⋊S3 kernel C3×C33⋊S3 C3×C3≀C3 C33⋊S3 C3≀C3 C3×He3 C3×3- 1+2 C34 He3 3- 1+2 C33 C32 C3 C1 # reps 1 1 2 2 1 2 1 2 4 2 12 3 6

Matrix representation of C3×C33⋊S3 in GL6(𝔽19)

 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7
,
 1 0 8 0 0 18 0 1 0 0 0 0 0 0 11 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 7
,
 1 8 1 18 0 11 0 11 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 1 0 0 0 0 0 0 11
,
 11 0 0 7 7 7 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7
,
 1 0 0 0 0 0 6 18 18 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0
,
 18 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 6 18 18 1 1 1 0 1 0 0 0 0 0 0 1 0 0 0

G:=sub<GL(6,GF(19))| [7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[1,0,0,0,0,0,0,1,0,0,0,0,8,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,18,0,0,0,0,7],[1,0,0,0,0,0,8,11,0,0,0,0,1,0,7,0,0,0,18,0,0,7,0,0,0,0,0,0,1,0,11,0,0,0,0,11],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,7,0,0,7,0,0,7,0,0,0,7,0,7,0,0,0,0,7],[1,6,0,0,0,0,0,18,1,0,0,0,0,18,0,0,0,0,0,1,0,0,0,1,0,1,0,1,0,0,0,1,0,0,1,0],[18,0,0,6,0,0,0,0,0,18,1,0,0,0,0,18,0,1,0,0,0,1,0,0,0,1,0,1,0,0,0,0,1,1,0,0] >;

C3×C33⋊S3 in GAP, Magma, Sage, TeX

C_3\times C_3^3\rtimes S_3
% in TeX

G:=Group("C3xC3^3:S3");
// GroupNames label

G:=SmallGroup(486,165);
// by ID

G=gap.SmallGroup(486,165);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,218,867,303,11344,1096,11669]);
// Polycyclic

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

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