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## G = C4×S3×A4order 288 = 25·32

### Direct product of C4, S3 and A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C4×S3×A4
 Chief series C1 — C3 — C2×C6 — C22×C6 — C6×A4 — C2×S3×A4 — C4×S3×A4
 Lower central C2×C6 — C4×S3×A4
 Upper central C1 — C4

Generators and relations for C4×S3×A4
G = < a,b,c,d,e,f | a4=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: 578 in 138 conjugacy classes, 33 normal (27 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, S3, C6, C6, C2×C4, C23, C23, C32, Dic3, Dic3, C12, C12, A4, A4, D6, D6, C2×C6, C2×C6, C22×C4, C22×C4, C24, C3×S3, C3×C6, C4×S3, C4×S3, C2×Dic3, C2×C12, C2×A4, C2×A4, C22×S3, C22×S3, C22×C6, C23×C4, C3×Dic3, C3×C12, C3×A4, S3×C6, C4×A4, C4×A4, S3×C2×C4, C22×Dic3, C22×C12, C22×A4, S3×C23, S3×C12, S3×A4, C6×A4, C2×C4×A4, S3×C22×C4, Dic3×A4, C12×A4, C2×S3×A4, C4×S3×A4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C12, A4, D6, C2×C6, C3×S3, C4×S3, C2×C12, C2×A4, S3×C6, C4×A4, C22×A4, S3×C12, S3×A4, C2×C4×A4, C2×S3×A4, C4×S3×A4

Smallest permutation representation of C4×S3×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 29 14)(2 30 15)(3 31 16)(4 32 13)(5 10 33)(6 11 34)(7 12 35)(8 9 36)(17 22 27)(18 23 28)(19 24 25)(20 21 26)
(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)(13 30)(14 31)(15 32)(16 29)(17 24)(18 21)(19 22)(20 23)(25 27)(26 28)(33 35)(34 36)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(29 31)(30 32)(33 35)(34 36)
(1 3)(2 4)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(1 26 35)(2 27 36)(3 28 33)(4 25 34)(5 31 18)(6 32 19)(7 29 20)(8 30 17)(9 15 22)(10 16 23)(11 13 24)(12 14 21)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L 12M 12N 12O 12P order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 6 6 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 size 1 1 3 3 3 3 9 9 2 4 4 8 8 1 1 3 3 3 3 9 9 2 4 4 6 6 8 8 12 12 12 12 2 2 4 4 4 4 6 6 8 8 8 8 12 12 12 12

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 6 6 6 type + + + + + + + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 D6 C3×S3 C4×S3 S3×C6 S3×C12 A4 C2×A4 C2×A4 C2×A4 C4×A4 S3×A4 C2×S3×A4 C4×S3×A4 kernel C4×S3×A4 Dic3×A4 C12×A4 C2×S3×A4 S3×C22×C4 S3×A4 C22×Dic3 C22×C12 S3×C23 C22×S3 C4×A4 C2×A4 C22×C4 A4 C23 C22 C4×S3 Dic3 C12 D6 S3 C4 C2 C1 # reps 1 1 1 1 2 4 2 2 2 8 1 1 2 2 2 4 1 1 1 1 4 1 1 2

Matrix representation of C4×S3×A4 in GL5(𝔽13)

 5 0 0 0 0 0 5 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 12 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 1 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 10 12 0 0 0 9 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 3 1 0 0 0 0 0 12
,
 3 0 0 0 0 0 3 0 0 0 0 0 3 0 8 0 0 0 0 1 0 0 0 4 10

G:=sub<GL(5,GF(13))| [5,0,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,1,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,10,9,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,3,0,0,0,0,1,0,0,0,0,0,12],[3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,8,1,10] >;

C4×S3×A4 in GAP, Magma, Sage, TeX

C_4\times S_3\times A_4
% in TeX

G:=Group("C4xS3xA4");
// GroupNames label

G:=SmallGroup(288,919);
// by ID

G=gap.SmallGroup(288,919);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-3,92,648,271,9414]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^4=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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