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## G = C4×C3⋊S4order 288 = 25·32

### Direct product of C4 and C3⋊S4

Series: Derived Chief Lower central Upper central

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

Generators and relations for C4×C3⋊S4
G = < a,b,c,d,e,f | a4=b3=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: 784 in 144 conjugacy classes, 29 normal (18 characteristic)
C1, C2, C2 [×4], C3, C3 [×3], C4, C4 [×5], C22, C22 [×6], S3 [×8], C6, C6 [×5], C2×C4 [×7], D4 [×4], C23, C23, C32, Dic3 [×7], C12, C12 [×4], A4 [×3], D6 [×7], C2×C6, C2×C6 [×2], C42, C22⋊C4 [×2], C4⋊C4, C22×C4, C22×C4, C2×D4, C3⋊S3 [×2], C3×C6, C4×S3 [×5], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12 [×2], S4 [×6], C2×A4 [×3], C22×S3, C22×C6, C4×D4, C3⋊Dic3, C3×C12, C3×A4, C2×C3⋊S3, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, A4⋊C4 [×3], C4×A4 [×3], S3×C2×C4, C2×C3⋊D4, C22×C12, C2×S4 [×3], C4×C3⋊S3, C3⋊S4 [×2], C6×A4, C4×C3⋊D4, C4×S4 [×3], C6.7S4, C12×A4, C2×C3⋊S4, C4×C3⋊S4
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D6 [×4], C3⋊S3, C4×S3 [×4], S4, C2×C3⋊S3, C2×S4, C4×C3⋊S3, C3⋊S4, C4×S4, C2×C3⋊S4, C4×C3⋊S4

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

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 4E ··· 4J 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D 12E ··· 12J order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 4 ··· 4 6 6 6 6 6 6 12 12 12 12 12 ··· 12 size 1 1 3 3 18 18 2 8 8 8 1 1 3 3 18 ··· 18 2 6 6 8 8 8 2 2 6 6 8 ··· 8

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 3 3 3 6 6 6 type + + + + + + + + + + + + image C1 C2 C2 C2 C4 S3 S3 D6 D6 C4×S3 C4×S3 S4 C2×S4 C4×S4 C3⋊S4 C2×C3⋊S4 C4×C3⋊S4 kernel C4×C3⋊S4 C6.7S4 C12×A4 C2×C3⋊S4 C3⋊S4 C4×A4 C22×C12 C2×A4 C22×C6 A4 C2×C6 C12 C6 C3 C4 C2 C1 # reps 1 1 1 1 4 3 1 3 1 6 2 2 2 4 1 1 2

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

 8 0 0 0 0 0 8 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 11 2 0 0 0 5 1 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 1 0 0 0 0 12 12 0 0 0 1 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 12 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 2 0 0 0 0 12 12 0 0 0 1 0
,
 2 11 0 0 0 8 11 0 0 0 0 0 12 11 0 0 0 0 1 0 0 0 0 12 12

G:=sub<GL(5,GF(13))| [8,0,0,0,0,0,8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[11,5,0,0,0,2,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,12,1,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,12,0,0,0,12,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,2,12,1,0,0,0,12,0],[2,8,0,0,0,11,11,0,0,0,0,0,12,0,0,0,0,11,1,12,0,0,0,0,12] >;

C4×C3⋊S4 in GAP, Magma, Sage, TeX

C_4\times C_3\rtimes S_4
% in TeX

G:=Group("C4xC3:S4");
// GroupNames label

G:=SmallGroup(288,908);
// by ID

G=gap.SmallGroup(288,908);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,36,451,1684,6053,782,3534,1350]);
// Polycyclic

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