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## G = C3×C4⋊S4order 288 = 25·32

### Direct product of C3 and C4⋊S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×A4 — C3×C4⋊S4
 Chief series C1 — C22 — A4 — C2×A4 — C6×A4 — C6×S4 — C3×C4⋊S4
 Lower central A4 — C2×A4 — C3×C4⋊S4
 Upper central C1 — C6 — C12

Generators and relations for C3×C4⋊S4
G = < a,b,c,d,e,f | a3=b4=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: 466 in 118 conjugacy classes, 24 normal (20 characteristic)
C1, C2, C2 [×4], C3, C3 [×2], C4, C4 [×3], C22, C22 [×8], S3 [×2], C6, C6 [×6], C2×C4 [×4], D4 [×6], C23, C23 [×2], C32, C12, C12 [×5], A4, A4, D6 [×2], C2×C6, C2×C6 [×8], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3×S3 [×2], C3×C6, D12, C2×C12 [×4], C3×D4 [×6], S4 [×2], C2×A4, C2×A4, C22×C6, C22×C6 [×2], C4⋊D4, C3×C12, C3×A4, S3×C6 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4, C4×A4, C4×A4, C22×C12, C6×D4 [×3], C2×S4 [×2], C3×D12, C3×S4 [×2], C6×A4, C3×C4⋊D4, C4⋊S4, C12×A4, C6×S4 [×2], C3×C4⋊S4
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, D6, C2×C6, C3×S3, D12, C3×D4, S4, S3×C6, C2×S4, C3×D12, C3×S4, C4⋊S4, C6×S4, C3×C4⋊S4

Smallest permutation representation of C3×C4⋊S4
On 36 points
Generators in S36
(1 35 23)(2 36 24)(3 33 21)(4 34 22)(5 12 31)(6 9 32)(7 10 29)(8 11 30)(13 19 27)(14 20 28)(15 17 25)(16 18 26)
(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)(2 4)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(33 35)(34 36)
(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(25 27)(26 28)(29 31)(30 32)
(1 13 5)(2 14 6)(3 15 7)(4 16 8)(9 36 20)(10 33 17)(11 34 18)(12 35 19)(21 25 29)(22 26 30)(23 27 31)(24 28 32)
(1 4)(2 3)(5 16)(6 15)(7 14)(8 13)(9 17)(10 20)(11 19)(12 18)(21 24)(22 23)(25 32)(26 31)(27 30)(28 29)(33 36)(34 35)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 6 6 type + + + + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D4 D6 C3×S3 C3×D4 D12 S3×C6 C3×D12 S4 C2×S4 C3×S4 C6×S4 C4⋊S4 C3×C4⋊S4 kernel C3×C4⋊S4 C12×A4 C6×S4 C4⋊S4 C4×A4 C2×S4 C22×C12 C3×A4 C22×C6 C22×C4 A4 C2×C6 C23 C22 C12 C6 C4 C2 C3 C1 # reps 1 1 2 2 2 4 1 1 1 2 2 2 2 4 2 2 4 4 1 2

Matrix representation of C3×C4⋊S4 in GL5(𝔽13)

 9 0 0 0 0 0 9 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 12 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 12 0 0 0 0 12 0 1 0 0 12 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 12 0 0 1 0 12 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 1 12 0 0 0 0 12 1
,
 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1

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

C3×C4⋊S4 in GAP, Magma, Sage, TeX

C_3\times C_4\rtimes S_4
% in TeX

G:=Group("C3xC4:S4");
// GroupNames label

G:=SmallGroup(288,898);
// by ID

G=gap.SmallGroup(288,898);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,2,197,92,1684,6053,285,3534,475]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^4=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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