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## G = C12×S4order 288 = 25·32

### Direct product of C12 and S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4 — C12×S4
 Chief series C1 — C22 — A4 — C2×A4 — C6×A4 — C6×S4 — C12×S4
 Lower central A4 — C12×S4
 Upper central C1 — C12

Generators and relations for C12×S4
G = < a,b,c,d,e | a12=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >

Subgroups: 374 in 118 conjugacy classes, 28 normal (24 characteristic)
C1, C2, C2 [×4], C3, C3 [×2], C4, C4 [×5], C22, C22 [×6], S3 [×2], C6, C6 [×6], C2×C4 [×7], D4 [×4], C23, C23, C32, Dic3, C12, C12 [×7], A4, A4, D6, C2×C6, C2×C6 [×6], C42, C22⋊C4 [×2], C4⋊C4, C22×C4, C22×C4, C2×D4, C3×S3 [×2], C3×C6, C4×S3, C2×C12 [×7], C3×D4 [×4], S4 [×2], C2×A4, C2×A4, C22×C6, C22×C6, C4×D4, C3×Dic3, C3×C12, C3×A4, S3×C6, C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, A4⋊C4, C4×A4, C4×A4, C22×C12, C22×C12, C6×D4, C2×S4, S3×C12, C3×S4 [×2], C6×A4, D4×C12, C4×S4, C3×A4⋊C4, C12×A4, C6×S4, C12×S4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, C12 [×2], D6, C2×C6, C3×S3, C4×S3, C2×C12, S4, S3×C6, C2×S4, S3×C12, C3×S4, C4×S4, C6×S4, C12×S4

Smallest permutation representation of C12×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 7)(2 8)(3 9)(4 10)(5 11)(6 12)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(1 34 20)(2 35 21)(3 36 22)(4 25 23)(5 26 24)(6 27 13)(7 28 14)(8 29 15)(9 30 16)(10 31 17)(11 32 18)(12 33 19)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 33)(14 34)(15 35)(16 36)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E ··· 4J 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 12A 12B 12C 12D 12E 12F 12G 12H 12I ··· 12T 12U ··· 12Z order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 ··· 4 6 6 6 6 6 6 6 6 6 6 6 6 6 12 12 12 12 12 12 12 12 12 ··· 12 12 ··· 12 size 1 1 3 3 6 6 1 1 8 8 8 1 1 3 3 6 ··· 6 1 1 3 3 3 3 6 6 6 6 8 8 8 1 1 1 1 3 3 3 3 6 ··· 6 8 ··· 8

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 type + + + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 D6 C3×S3 C4×S3 S3×C6 S3×C12 S4 C2×S4 C3×S4 C4×S4 C6×S4 C12×S4 kernel C12×S4 C3×A4⋊C4 C12×A4 C6×S4 C4×S4 C3×S4 A4⋊C4 C4×A4 C2×S4 S4 C22×C12 C22×C6 C22×C4 C2×C6 C23 C22 C12 C6 C4 C3 C2 C1 # reps 1 1 1 1 2 4 2 2 2 8 1 1 2 2 2 4 2 2 4 4 4 8

Matrix representation of C12×S4 in GL3(𝔽13) generated by

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

C12×S4 in GAP, Magma, Sage, TeX

C_{12}\times S_4
% in TeX

G:=Group("C12xS4");
// GroupNames label

G:=SmallGroup(288,897);
// by ID

G=gap.SmallGroup(288,897);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,2,92,1684,6053,285,3534,475]);
// Polycyclic

G:=Group<a,b,c,d,e|a^12=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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