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## G = C2×C3≀S3order 324 = 22·34

### Direct product of C2 and C3≀S3

Aliases: C2×C3≀S3, C337D6, He3⋊(C2×C6), (C2×He3)⋊C6, C3≀C33C22, (C32×C6)⋊1S3, He3⋊C21C6, C32.1(S3×C6), C6.15(C32⋊C6), (C2×C3≀C3)⋊2C2, (C3×C6).4(C3×S3), C3.6(C2×C32⋊C6), (C2×He3⋊C2)⋊1C3, SmallGroup(324,68)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — C2×C3≀S3
 Chief series C1 — C3 — C32 — He3 — C3≀C3 — C3≀S3 — C2×C3≀S3
 Lower central He3 — C2×C3≀S3
 Upper central C1 — C6

Generators and relations for C2×C3≀S3
G = < a,b,c,d,e | a2=b3=c3=d3=e6=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe-1=b-1c, cd=dc, ce=ec, ede-1=b-1d-1 >

Subgroups: 376 in 84 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2, C3, C3, C22, S3, C6, C6, C9, C32, C32, D6, C2×C6, C18, C3×S3, C3×C6, C3×C6, He3, 3- 1+2, C33, S3×C6, C62, He3⋊C2, C2×He3, C2×3- 1+2, S3×C32, C32×C6, C3≀C3, C2×He3⋊C2, S3×C3×C6, C3≀S3, C2×C3≀C3, C2×C3≀S3
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, S3×C6, C32⋊C6, C2×C32⋊C6, C3≀S3, C2×C3≀S3

Permutation representations of C2×C3≀S3
On 18 points - transitive group 18T119
Generators in S18
(1 2)(3 4)(5 6)(7 15)(8 16)(9 17)(10 18)(11 13)(12 14)
(1 6 4)(2 5 3)(8 10 12)(14 16 18)
(1 4 6)(2 3 5)(7 9 11)(8 10 12)(13 15 17)(14 16 18)
(1 16 11)(2 8 13)(3 10 15)(4 18 7)(5 12 17)(6 14 9)
(1 2)(3 4)(5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)

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

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

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

G:=TransitiveGroup(18,119);

44 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 3I 3J 6A 6B 6C ··· 6H 6I 6J ··· 6Y 6Z 9A 9B 18A 18B order 1 2 2 2 3 3 3 ··· 3 3 3 6 6 6 ··· 6 6 6 ··· 6 6 9 9 18 18 size 1 1 9 9 1 1 3 ··· 3 6 18 1 1 3 ··· 3 6 9 ··· 9 18 18 18 18 18

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 6 6 type + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 C3≀S3 C2×C3≀S3 C32⋊C6 C2×C32⋊C6 kernel C2×C3≀S3 C3≀S3 C2×C3≀C3 C2×He3⋊C2 He3⋊C2 C2×He3 C32×C6 C33 C3×C6 C32 C2 C1 C6 C3 # reps 1 2 1 2 4 2 1 1 2 2 12 12 1 1

Matrix representation of C2×C3≀S3 in GL3(𝔽7) generated by

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

C2×C3≀S3 in GAP, Magma, Sage, TeX

C_2\times C_3\wr S_3
% in TeX

G:=Group("C2xC3wrS3");
// GroupNames label

G:=SmallGroup(324,68);
// by ID

G=gap.SmallGroup(324,68);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,579,303,5404,382]);
// Polycyclic

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

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