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## G = C3×C32⋊4D6order 324 = 22·34

### Direct product of C3 and C32⋊4D6

Aliases: C3×C324D6, C3C22, C3314D6, C343C22, C329S32, C338(C2×C6), C327(S3×C6), C32(C3×S32), (C3×C3⋊S3)⋊3S3, (C3×C3⋊S3)⋊5C6, C3⋊S33(C3×S3), (C32×C3⋊S3)⋊3C2, SmallGroup(324,167)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C3×C32⋊4D6
 Chief series C1 — C3 — C32 — C33 — C34 — C32×C3⋊S3 — C3×C32⋊4D6
 Lower central C33 — C3×C32⋊4D6
 Upper central C1 — C3

Generators and relations for C3×C324D6
G = < a,b,c,d,e | a3=b3=c3=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 736 in 174 conjugacy classes, 30 normal (6 characteristic)
C1, C2 [×3], C3, C3 [×3], C3 [×12], C22, S3 [×9], C6 [×9], C32 [×6], C32 [×40], D6 [×3], C2×C6, C3×S3 [×30], C3⋊S3 [×3], C3×C6 [×3], C33, C33 [×3], C33 [×12], S32 [×3], S3×C6 [×3], S3×C32 [×9], C3×C3⋊S3 [×6], C3×C3⋊S3 [×3], C34, C3×S32 [×3], C324D6, C32×C3⋊S3 [×3], C3×C324D6
Quotients: C1, C2 [×3], C3, C22, S3 [×3], C6 [×3], D6 [×3], C2×C6, C3×S3 [×3], S32 [×3], S3×C6 [×3], C3×S32 [×3], C324D6, C3×C324D6

Permutation representations of C3×C324D6
On 12 points - transitive group 12T130
Generators in S12
(1 5 3)(2 6 4)(7 9 11)(8 10 12)
(1 5 3)(2 4 6)(7 11 9)(8 10 12)
(1 3 5)(2 6 4)(7 11 9)(8 10 12)
(1 2 3 4 5 6)(7 8 9 10 11 12)
(1 10)(2 9)(3 8)(4 7)(5 12)(6 11)

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

G:=Group( (1,5,3)(2,6,4)(7,9,11)(8,10,12), (1,5,3)(2,4,6)(7,11,9)(8,10,12), (1,3,5)(2,6,4)(7,11,9)(8,10,12), (1,2,3,4,5,6)(7,8,9,10,11,12), (1,10)(2,9)(3,8)(4,7)(5,12)(6,11) );

G=PermutationGroup([(1,5,3),(2,6,4),(7,9,11),(8,10,12)], [(1,5,3),(2,4,6),(7,11,9),(8,10,12)], [(1,3,5),(2,6,4),(7,11,9),(8,10,12)], [(1,2,3,4,5,6),(7,8,9,10,11,12)], [(1,10),(2,9),(3,8),(4,7),(5,12),(6,11)])

G:=TransitiveGroup(12,130);

On 18 points - transitive group 18T120
Generators in S18
(1 17 12)(2 18 7)(3 13 8)(4 14 9)(5 15 10)(6 16 11)
(1 3 5)(2 6 4)(7 11 9)(8 10 12)(13 15 17)(14 18 16)
(1 17 12)(2 7 18)(3 13 8)(4 9 14)(5 15 10)(6 11 16)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 6)(2 5)(3 4)(7 10)(8 9)(11 12)(13 14)(15 18)(16 17)

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

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

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

G:=TransitiveGroup(18,120);

Polynomial with Galois group C3×C324D6 over ℚ
actionf(x)Disc(f)
12T130x12-x9+5x6-8x3+4222·318·56

45 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3K 3L ··· 3Z 6A ··· 6F 6G ··· 6O order 1 2 2 2 3 3 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 size 1 9 9 9 1 1 2 ··· 2 4 ··· 4 9 ··· 9 18 ··· 18

45 irreducible representations

 dim 1 1 1 1 2 2 2 2 4 4 4 4 type + + + + + image C1 C2 C3 C6 S3 D6 C3×S3 S3×C6 S32 C3×S32 C32⋊4D6 C3×C32⋊4D6 kernel C3×C32⋊4D6 C32×C3⋊S3 C32⋊4D6 C3×C3⋊S3 C3×C3⋊S3 C33 C3⋊S3 C32 C32 C3 C3 C1 # reps 1 3 2 6 3 3 6 6 3 6 2 4

Matrix representation of C3×C324D6 in GL4(𝔽7) generated by

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 5 3 2 3 1 3 3 0 4 4 0 6 0 0 0 4
,
 3 6 3 2 6 3 4 2 0 0 2 0 0 0 0 4
,
 1 5 6 6 3 2 5 2 2 5 6 3 1 1 6 5
,
 6 0 0 0 1 1 0 0 6 1 4 1 2 2 6 3
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[5,1,4,0,3,3,4,0,2,3,0,0,3,0,6,4],[3,6,0,0,6,3,0,0,3,4,2,0,2,2,0,4],[1,3,2,1,5,2,5,1,6,5,6,6,6,2,3,5],[6,1,6,2,0,1,1,2,0,0,4,6,0,0,1,3] >;

C3×C324D6 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_4D_6
% in TeX

G:=Group("C3xC3^2:4D6");
// GroupNames label

G:=SmallGroup(324,167);
// by ID

G=gap.SmallGroup(324,167);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,579,297,1090,7781]);
// Polycyclic

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

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