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## G = C2×C6×S4order 288 = 25·32

### Direct product of C2×C6 and S4

Aliases: C2×C6×S4, C23⋊(S3×C6), A4⋊(C22×C6), C244(C3×S3), (C23×C6)⋊2S3, (C22×C6)⋊2D6, (C6×A4)⋊2C22, (C22×A4)⋊5C6, (C3×A4)⋊2C23, C22⋊(S3×C2×C6), (C2×A4)⋊(C2×C6), (A4×C2×C6)⋊5C2, (C2×C6)⋊2(C22×S3), SmallGroup(288,1033)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C6×S4
G = < a,b,c,d,e,f | a2=b6=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 890 in 272 conjugacy classes, 52 normal (14 characteristic)
C1, C2 [×3], C2 [×8], C3, C3 [×2], C4 [×4], C22 [×2], C22 [×26], S3 [×4], C6 [×3], C6 [×14], C2×C4 [×6], D4 [×16], C23 [×3], C23 [×14], C32, C12 [×4], A4, A4, D6 [×6], C2×C6 [×2], C2×C6 [×28], C22×C4, C2×D4 [×12], C24, C24, C3×S3 [×4], C3×C6 [×3], C2×C12 [×6], C3×D4 [×16], S4 [×4], C2×A4 [×3], C2×A4 [×3], C22×S3, C22×C6 [×3], C22×C6 [×14], C22×D4, C3×A4, S3×C6 [×6], C62, C22×C12, C6×D4 [×12], C2×S4 [×6], C22×A4, C22×A4, C23×C6, C23×C6, C3×S4 [×4], C6×A4 [×3], S3×C2×C6, D4×C2×C6, C22×S4, C6×S4 [×6], A4×C2×C6, C2×C6×S4
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], C23, D6 [×3], C2×C6 [×7], C3×S3, S4, C22×S3, C22×C6, S3×C6 [×3], C2×S4 [×3], C3×S4, S3×C2×C6, C22×S4, C6×S4 [×3], C2×C6×S4

Smallest permutation representation of C2×C6×S4
On 36 points
Generators in S36
(1 24)(2 19)(3 20)(4 21)(5 22)(6 23)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 31)(26 32)(27 33)(28 34)(29 35)(30 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 24)(2 19)(3 20)(4 21)(5 22)(6 23)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)
(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(1 12 32)(2 7 33)(3 8 34)(4 9 35)(5 10 36)(6 11 31)(13 27 19)(14 28 20)(15 29 21)(16 30 22)(17 25 23)(18 26 24)
(1 24)(2 19)(3 20)(4 21)(5 22)(6 23)(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(13 33)(14 34)(15 35)(16 36)(17 31)(18 32)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A ··· 6F 6G ··· 6N 6O ··· 6V 6W ··· 6AE 12A ··· 12H order 1 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 3 3 3 3 6 6 6 6 1 1 8 8 8 6 6 6 6 1 ··· 1 3 ··· 3 6 ··· 6 8 ··· 8 6 ··· 6

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 3 type + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 S4 C2×S4 C3×S4 C6×S4 kernel C2×C6×S4 C6×S4 A4×C2×C6 C22×S4 C2×S4 C22×A4 C23×C6 C22×C6 C24 C23 C2×C6 C6 C22 C2 # reps 1 6 1 2 12 2 1 3 2 6 2 6 4 12

Matrix representation of C2×C6×S4 in GL5(𝔽13)

 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 4 0 0 0 0 0 4 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 1 1 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 12 0 12
,
 0 12 0 0 0 1 12 0 0 0 0 0 12 12 11 0 0 1 0 0 0 0 0 0 1
,
 12 1 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 1 1 2 0 0 0 0 12

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,1,0,0,0,12,1,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,12,0,0,0,12,0,0,0,0,0,12],[0,1,0,0,0,12,12,0,0,0,0,0,12,1,0,0,0,12,0,0,0,0,11,0,1],[12,0,0,0,0,1,1,0,0,0,0,0,12,1,0,0,0,0,1,0,0,0,0,2,12] >;

C2×C6×S4 in GAP, Magma, Sage, TeX

C_2\times C_6\times S_4
% in TeX

G:=Group("C2xC6xS4");
// GroupNames label

G:=SmallGroup(288,1033);
// by ID

G=gap.SmallGroup(288,1033);
# by ID

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

G:=Group<a,b,c,d,e,f|a^2=b^6=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,b*f=f*b,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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