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## G = C4×S3≀C2order 288 = 25·32

### Direct product of C4 and S3≀C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3 — C4×S3≀C2
 Chief series C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C2×S32 — C2×S3≀C2 — C4×S3≀C2
 Lower central C32 — C3⋊S3 — C4×S3≀C2
 Upper central C1 — C4

Generators and relations for C4×S3≀C2
G = < a,b,c,d,e | a4=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=c, ebe=dcd-1=b-1, ce=ec, ede=d-1 >

Subgroups: 736 in 148 conjugacy classes, 35 normal (17 characteristic)
C1, C2, C2 [×6], C3 [×2], C4, C4 [×6], C22 [×9], S3 [×8], C6 [×6], C2×C4 [×9], D4 [×4], C23 [×2], C32, Dic3 [×4], C12 [×4], D6 [×12], C2×C6 [×2], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C3×S3 [×4], C3⋊S3 [×2], C3×C6, C4×S3 [×8], C2×Dic3 [×2], C2×C12 [×2], C22×S3 [×2], C4×D4, C3×Dic3 [×2], C3⋊Dic3, C3×C12, C32⋊C4 [×2], C32⋊C4, S32 [×4], S32 [×2], S3×C6 [×2], C2×C3⋊S3, S3×C2×C4 [×2], S3×Dic3 [×2], C6.D6 [×2], S3×C12 [×2], C4×C3⋊S3, S3≀C2 [×4], C2×C32⋊C4 [×2], C2×S32 [×2], S32⋊C4 [×2], C3⋊S3.Q8, C4×C32⋊C4, C4×S32 [×2], C2×S3≀C2, C4×S3≀C2
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, C22×C4, C2×D4, C4○D4, C4×D4, S3≀C2, C2×S3≀C2, C4×S3≀C2

Permutation representations of C4×S3≀C2
On 24 points - transitive group 24T643
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(5 10 23)(6 11 24)(7 12 21)(8 9 22)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)
(1 24)(2 21)(3 22)(4 23)(5 20 10 15)(6 17 11 16)(7 18 12 13)(8 19 9 14)
(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)

G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,16,17)(2,13,18)(3,14,19)(4,15,20), (1,24)(2,21)(3,22)(4,23)(5,20,10,15)(6,17,11,16)(7,18,12,13)(8,19,9,14), (1,3)(2,4)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)>;

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), (5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,16,17)(2,13,18)(3,14,19)(4,15,20), (1,24)(2,21)(3,22)(4,23)(5,20,10,15)(6,17,11,16)(7,18,12,13)(8,19,9,14), (1,3)(2,4)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24) );

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)], [(5,10,23),(6,11,24),(7,12,21),(8,9,22)], [(1,16,17),(2,13,18),(3,14,19),(4,15,20)], [(1,24),(2,21),(3,22),(4,23),(5,20,10,15),(6,17,11,16),(7,18,12,13),(8,19,9,14)], [(1,3),(2,4),(5,12),(6,9),(7,10),(8,11),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)])

G:=TransitiveGroup(24,643);

On 24 points - transitive group 24T650
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 23 10)(6 24 11)(7 21 12)(8 22 9)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 10 23)(6 11 24)(7 12 21)(8 9 22)
(1 22 3 24)(2 23 4 21)(5 15 12 18)(6 16 9 19)(7 13 10 20)(8 14 11 17)
(1 24)(2 21)(3 22)(4 23)(5 15)(6 16)(7 13)(8 14)(9 19)(10 20)(11 17)(12 18)

G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,22,3,24)(2,23,4,21)(5,15,12,18)(6,16,9,19)(7,13,10,20)(8,14,11,17), (1,24)(2,21)(3,22)(4,23)(5,15)(6,16)(7,13)(8,14)(9,19)(10,20)(11,17)(12,18)>;

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), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,22,3,24)(2,23,4,21)(5,15,12,18)(6,16,9,19)(7,13,10,20)(8,14,11,17), (1,24)(2,21)(3,22)(4,23)(5,15)(6,16)(7,13)(8,14)(9,19)(10,20)(11,17)(12,18) );

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)], [(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,23,10),(6,24,11),(7,21,12),(8,22,9)], [(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,10,23),(6,11,24),(7,12,21),(8,9,22)], [(1,22,3,24),(2,23,4,21),(5,15,12,18),(6,16,9,19),(7,13,10,20),(8,14,11,17)], [(1,24),(2,21),(3,22),(4,23),(5,15),(6,16),(7,13),(8,14),(9,19),(10,20),(11,17),(12,18)])

G:=TransitiveGroup(24,650);

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D 12E 12F 12G 12H order 1 2 2 2 2 2 2 2 3 3 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 6 12 12 12 12 12 12 12 12 size 1 1 6 6 6 6 9 9 4 4 1 1 6 6 6 6 9 9 18 18 18 18 4 4 12 12 12 12 4 4 4 4 12 12 12 12

36 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 4 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 D4 D4 C4○D4 S3≀C2 C2×S3≀C2 C4×S3≀C2 kernel C4×S3≀C2 S32⋊C4 C3⋊S3.Q8 C4×C32⋊C4 C4×S32 C2×S3≀C2 S3≀C2 C3⋊Dic3 C3×C12 C3⋊S3 C4 C2 C1 # reps 1 2 1 1 2 1 8 1 1 2 4 4 8

Matrix representation of C4×S3≀C2 in GL4(𝔽5) generated by

 3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3
,
 0 0 0 1 0 0 2 0 0 2 4 0 4 0 0 4
,
 4 0 0 4 0 0 2 0 0 2 4 0 1 0 0 0
,
 0 0 2 0 4 0 0 4 0 0 0 3 0 4 3 0
,
 0 1 2 0 1 0 0 1 0 0 0 2 0 0 3 0
G:=sub<GL(4,GF(5))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[0,0,0,4,0,0,2,0,0,2,4,0,1,0,0,4],[4,0,0,1,0,0,2,0,0,2,4,0,4,0,0,0],[0,4,0,0,0,0,0,4,2,0,0,3,0,4,3,0],[0,1,0,0,1,0,0,0,2,0,0,3,0,1,2,0] >;

C4×S3≀C2 in GAP, Magma, Sage, TeX

C_4\times S_3\wr C_2
% in TeX

G:=Group("C4xS3wrC2");
// GroupNames label

G:=SmallGroup(288,877);
// by ID

G=gap.SmallGroup(288,877);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,100,2693,2028,362,797,1203]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^3=c^3=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=c,e*b*e=d*c*d^-1=b^-1,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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