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## G = C6×C3⋊S4order 432 = 24·33

### Direct product of C6 and C3⋊S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×A4 — C6×C3⋊S4
 Chief series C1 — C22 — C2×C6 — C3×A4 — C32×A4 — C3×C3⋊S4 — C6×C3⋊S4
 Lower central C3×A4 — C6×C3⋊S4
 Upper central C1 — C6

Generators and relations for C6×C3⋊S4
G = < a,b,c,d,e,f | a6=b3=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 1072 in 208 conjugacy classes, 38 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, C32, Dic3, C12, A4, A4, D6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C2×C12, C3×D4, S4, C2×A4, C2×A4, C22×S3, C22×C6, C22×C6, C33, C3×Dic3, C3×A4, C3×A4, C3×A4, S3×C6, C2×C3⋊S3, C62, C62, C2×C3⋊D4, C6×D4, C2×S4, C3×C3⋊S3, C32×C6, C6×Dic3, C3×C3⋊D4, C3×S4, C3⋊S4, C6×A4, C6×A4, C6×A4, S3×C2×C6, C2×C62, C32×A4, C6×C3⋊S3, C6×C3⋊D4, C6×S4, C2×C3⋊S4, C3×C3⋊S4, A4×C3×C6, C6×C3⋊S4
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, C3⋊S3, S4, S3×C6, C2×C3⋊S3, C2×S4, C3×C3⋊S3, C3×S4, C3⋊S4, C6×C3⋊S3, C6×S4, C2×C3⋊S4, C3×C3⋊S4, C6×C3⋊S4

Smallest permutation representation of C6×C3⋊S4
On 36 points
Generators in S36
(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)
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 33 35)(32 34 36)
(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)
(1 4)(2 5)(3 6)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(31 34)(32 35)(33 36)
(1 11 17)(2 12 18)(3 7 13)(4 8 14)(5 9 15)(6 10 16)(19 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 19)(7 28)(8 29)(9 30)(10 25)(11 26)(12 27)(13 34)(14 35)(15 36)(16 31)(17 32)(18 33)

G:=sub<Sym(36)| (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), (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,33,35)(32,34,36), (7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30), (1,4)(2,5)(3,6)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(31,34)(32,35)(33,36), (1,11,17)(2,12,18)(3,7,13)(4,8,14)(5,9,15)(6,10,16)(19,31,25)(20,32,26)(21,33,27)(22,34,28)(23,35,29)(24,36,30), (1,20)(2,21)(3,22)(4,23)(5,24)(6,19)(7,28)(8,29)(9,30)(10,25)(11,26)(12,27)(13,34)(14,35)(15,36)(16,31)(17,32)(18,33)>;

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)(25,26,27,28,29,30)(31,32,33,34,35,36), (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,33,35)(32,34,36), (7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30), (1,4)(2,5)(3,6)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(31,34)(32,35)(33,36), (1,11,17)(2,12,18)(3,7,13)(4,8,14)(5,9,15)(6,10,16)(19,31,25)(20,32,26)(21,33,27)(22,34,28)(23,35,29)(24,36,30), (1,20)(2,21)(3,22)(4,23)(5,24)(6,19)(7,28)(8,29)(9,30)(10,25)(11,26)(12,27)(13,34)(14,35)(15,36)(16,31)(17,32)(18,33) );

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),(25,26,27,28,29,30),(31,32,33,34,35,36)], [(1,5,3),(2,6,4),(7,11,9),(8,12,10),(13,17,15),(14,18,16),(19,21,23),(20,22,24),(25,27,29),(26,28,30),(31,33,35),(32,34,36)], [(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(25,28),(26,29),(27,30)], [(1,4),(2,5),(3,6),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(31,34),(32,35),(33,36)], [(1,11,17),(2,12,18),(3,7,13),(4,8,14),(5,9,15),(6,10,16),(19,31,25),(20,32,26),(21,33,27),(22,34,28),(23,35,29),(24,36,30)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,19),(7,28),(8,29),(9,30),(10,25),(11,26),(12,27),(13,34),(14,35),(15,36),(16,31),(17,32),(18,33)]])

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 3F ··· 3N 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J ··· 6O 6P ··· 6X 6Y 6Z 6AA 6AB 12A 12B 12C 12D order 1 2 2 2 2 2 3 3 3 3 3 3 ··· 3 4 4 6 6 6 6 6 6 6 6 6 6 ··· 6 6 ··· 6 6 6 6 6 12 12 12 12 size 1 1 3 3 18 18 1 1 2 2 2 8 ··· 8 18 18 1 1 2 2 2 3 3 3 3 6 ··· 6 8 ··· 8 18 18 18 18 18 18 18 18

54 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 6 6 6 6 type + + + + + + + + + + + image C1 C2 C2 C3 C6 C6 S3 S3 D6 D6 C3×S3 C3×S3 S3×C6 S3×C6 S4 C2×S4 C3×S4 C6×S4 C3⋊S4 C2×C3⋊S4 C3×C3⋊S4 C6×C3⋊S4 kernel C6×C3⋊S4 C3×C3⋊S4 A4×C3×C6 C2×C3⋊S4 C3⋊S4 C6×A4 C6×A4 C2×C62 C3×A4 C62 C2×A4 C22×C6 A4 C2×C6 C3×C6 C32 C6 C3 C6 C3 C2 C1 # reps 1 2 1 2 4 2 3 1 3 1 6 2 6 2 2 2 4 4 1 1 2 2

Matrix representation of C6×C3⋊S4 in GL7(𝔽13)

 4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 3 0 0 0 0 0 0 3 9 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12
,
 9 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 9 0 0 0 0 0 0 10 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0
,
 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0

G:=sub<GL(7,GF(13))| [4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12],[9,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,9,10,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0] >;

C6×C3⋊S4 in GAP, Magma, Sage, TeX

C_6\times C_3\rtimes S_4
% in TeX

G:=Group("C6xC3:S4");
// GroupNames label

G:=SmallGroup(432,761);
// by ID

G=gap.SmallGroup(432,761);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,675,2524,9077,782,5298,1350]);
// Polycyclic

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

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