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## G = S3×C6×A4order 432 = 24·33

### Direct product of C6, S3 and A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — S3×C6×A4
 Chief series C1 — C3 — C2×C6 — C62 — C32×A4 — C3×S3×A4 — S3×C6×A4
 Lower central C2×C6 — S3×C6×A4
 Upper central C1 — C6

Generators and relations for S3×C6×A4
G = < a,b,c,d,e,f | a6=b3=c2=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 964 in 232 conjugacy classes, 56 normal (25 characteristic)
C1, C2, C2, C3, C3, C22, C22, S3, S3, C6, C6, C23, C23, C32, C32, A4, A4, D6, D6, C2×C6, C2×C6, C24, C3×S3, C3×S3, C3×C6, C3×C6, C2×A4, C2×A4, C22×S3, C22×S3, C22×C6, C22×C6, C33, C3×A4, C3×A4, C3×A4, S3×C6, S3×C6, C62, C62, C22×A4, S3×C23, C23×C6, S3×C32, C32×C6, S3×A4, C6×A4, C6×A4, C6×A4, S3×C2×C6, S3×C2×C6, C2×C62, C32×A4, S3×C3×C6, C2×S3×A4, A4×C2×C6, S3×C22×C6, C3×S3×A4, A4×C3×C6, S3×C6×A4
Quotients: C1, C2, C3, C22, S3, C6, C32, A4, D6, C2×C6, C3×S3, C3×C6, C2×A4, C3×A4, S3×C6, C62, C22×A4, S3×C32, S3×A4, C6×A4, S3×C3×C6, C2×S3×A4, A4×C2×C6, C3×S3×A4, S3×C6×A4

Smallest permutation representation of S3×C6×A4
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 3 5)(2 4 6)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 35 33)(32 36 34)
(1 23)(2 24)(3 19)(4 20)(5 21)(6 22)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)
(1 4)(2 5)(3 6)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(31 34)(32 35)(33 36)
(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)
(1 17 11)(2 18 12)(3 13 7)(4 14 8)(5 15 9)(6 16 10)(19 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 3F ··· 3K 3L ··· 3Q 6A 6B 6C 6D 6E 6F ··· 6M 6N ··· 6S 6T ··· 6Y 6Z ··· 6AE 6AF 6AG 6AH 6AI 6AJ ··· 6AU order 1 2 2 2 2 2 2 2 3 3 3 3 3 3 ··· 3 3 ··· 3 6 6 6 6 6 6 ··· 6 6 ··· 6 6 ··· 6 6 ··· 6 6 6 6 6 6 ··· 6 size 1 1 3 3 3 3 9 9 1 1 2 2 2 4 ··· 4 8 ··· 8 1 1 2 2 2 3 ··· 3 4 ··· 4 6 ··· 6 8 ··· 8 9 9 9 9 12 ··· 12

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 6 6 6 6 type + + + + + + + + + + image C1 C2 C2 C3 C3 C6 C6 C6 C6 S3 D6 C3×S3 C3×S3 S3×C6 S3×C6 A4 C2×A4 C2×A4 C3×A4 C6×A4 C6×A4 S3×A4 C2×S3×A4 C3×S3×A4 S3×C6×A4 kernel S3×C6×A4 C3×S3×A4 A4×C3×C6 C2×S3×A4 S3×C22×C6 S3×A4 C6×A4 S3×C2×C6 C2×C62 C6×A4 C3×A4 C2×A4 C22×C6 A4 C2×C6 S3×C6 C3×S3 C3×C6 D6 S3 C6 C6 C3 C2 C1 # reps 1 2 1 6 2 12 6 4 2 1 1 6 2 6 2 1 2 1 2 4 2 1 1 2 2

Matrix representation of S3×C6×A4 in GL5(𝔽7)

 6 0 0 0 0 0 6 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 0 0 2
,
 4 0 0 0 0 2 2 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 6 1 0 0 0 0 1 0 0 0 0 0 6 0 0 0 0 0 6 0 0 0 0 0 6
,
 1 0 0 0 0 0 1 0 0 0 0 0 6 0 0 0 0 0 1 0 0 0 0 0 6
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 6 0 0 0 0 0 6
,
 4 0 0 0 0 0 4 0 0 0 0 0 0 2 0 0 0 0 0 2 0 0 2 0 0

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

S3×C6×A4 in GAP, Magma, Sage, TeX

S_3\times C_6\times A_4
% in TeX

G:=Group("S3xC6xA4");
// GroupNames label

G:=SmallGroup(432,763);
// by ID

G=gap.SmallGroup(432,763);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,2,-3,963,397,14118]);
// Polycyclic

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

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