Copied to
clipboard

G = C2×C3≀S3order 324 = 22·34

Direct product of C2 and C3≀S3

direct product, non-abelian, supersoluble, monomial

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

C1C3He3 — C2×C3≀S3
C1C3C32He3C3≀C3C3≀S3 — C2×C3≀S3
He3 — C2×C3≀S3
C1C6

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 [×2], C3, C3 [×5], C22, S3 [×4], C6, C6 [×13], C9, C32, C32 [×5], D6 [×2], C2×C6 [×4], C18, C3×S3 [×10], C3×C6, C3×C6 [×7], He3, 3- 1+2, C33, S3×C6 [×5], C62, He3⋊C2 [×2], C2×He3, C2×3- 1+2, S3×C32 [×2], C32×C6, C3≀C3, C2×He3⋊C2, S3×C3×C6, C3≀S3 [×2], C2×C3≀C3, C2×C3≀S3
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], 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 16)(8 17)(9 18)(10 13)(11 14)(12 15)
(1 6 3)(2 5 4)(7 9 11)(14 16 18)
(1 3 6)(2 4 5)(7 9 11)(8 10 12)(13 15 17)(14 16 18)
(1 18 12)(2 9 15)(3 14 8)(4 11 17)(5 7 13)(6 16 10)
(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)(7,16)(8,17)(9,18)(10,13)(11,14)(12,15), (1,6,3)(2,5,4)(7,9,11)(14,16,18), (1,3,6)(2,4,5)(7,9,11)(8,10,12)(13,15,17)(14,16,18), (1,18,12)(2,9,15)(3,14,8)(4,11,17)(5,7,13)(6,16,10), (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)(7,16)(8,17)(9,18)(10,13)(11,14)(12,15), (1,6,3)(2,5,4)(7,9,11)(14,16,18), (1,3,6)(2,4,5)(7,9,11)(8,10,12)(13,15,17)(14,16,18), (1,18,12)(2,9,15)(3,14,8)(4,11,17)(5,7,13)(6,16,10), (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),(7,16),(8,17),(9,18),(10,13),(11,14),(12,15)], [(1,6,3),(2,5,4),(7,9,11),(14,16,18)], [(1,3,6),(2,4,5),(7,9,11),(8,10,12),(13,15,17),(14,16,18)], [(1,18,12),(2,9,15),(3,14,8),(4,11,17),(5,7,13),(6,16,10)], [(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 2A2B2C3A3B3C···3H3I3J6A6B6C···6H6I6J···6Y6Z9A9B18A18B
order1222333···333666···666···66991818
size1199113···3618113···369···91818181818

44 irreducible representations

dim11111122223366
type+++++++
imageC1C2C2C3C6C6S3D6C3×S3S3×C6C3≀S3C2×C3≀S3C32⋊C6C2×C32⋊C6
kernelC2×C3≀S3C3≀S3C2×C3≀C3C2×He3⋊C2He3⋊C2C2×He3C32×C6C33C3×C6C32C2C1C6C3
# reps1212421122121211

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

600
060
006
,
015
102
620
,
400
040
004
,
400
204
133
,
150
023
010
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

׿
×
𝔽