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## G = C8×S4order 192 = 26·3

### Direct product of C8 and S4

Aliases: C8×S4, C22⋊(S3×C8), A4⋊C86C2, (C8×A4)⋊4C2, A41(C2×C8), C2.1(C4×S4), A4⋊C4.2C4, (C4×S4).3C2, (C2×S4).2C4, C4.26(C2×S4), (C22×C8)⋊1S3, C23.1(C4×S3), (C22×C4).8D6, (C4×A4).12C22, (C2×A4).1(C2×C4), SmallGroup(192,958)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4 — C8×S4
 Chief series C1 — C22 — A4 — C2×A4 — C4×A4 — C4×S4 — C8×S4
 Lower central A4 — C8×S4
 Upper central C1 — C8

Generators and relations for C8×S4
G = < a,b,c,d,e | a8=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: 246 in 79 conjugacy classes, 19 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C8, C2×C4, D4, C23, C23, Dic3, C12, A4, D6, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C3⋊C8, C24, C4×S3, S4, C2×A4, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C22×C8, S3×C8, A4⋊C4, C4×A4, C2×S4, C8×D4, A4⋊C8, C8×A4, C4×S4, C8×S4
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D6, C2×C8, C4×S3, S4, S3×C8, C2×S4, C4×S4, C8×S4

Permutation representations of C8×S4
On 24 points - transitive group 24T322
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 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)
(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)
(1 11 19)(2 12 20)(3 13 21)(4 14 22)(5 15 23)(6 16 24)(7 9 17)(8 10 18)
(1 5)(2 6)(3 7)(4 8)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)

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

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

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

G:=TransitiveGroup(24,322);

40 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E ··· 4J 6 8A 8B 8C 8D 8E 8F 8G 8H 8I ··· 8P 12A 12B 24A 24B 24C 24D order 1 2 2 2 2 2 3 4 4 4 4 4 ··· 4 6 8 8 8 8 8 8 8 8 8 ··· 8 12 12 24 24 24 24 size 1 1 3 3 6 6 8 1 1 3 3 6 ··· 6 8 1 1 1 1 3 3 3 3 6 ··· 6 8 8 8 8 8 8

40 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 type + + + + + + + + image C1 C2 C2 C2 C4 C4 C8 S3 D6 C4×S3 S3×C8 S4 C2×S4 C4×S4 C8×S4 kernel C8×S4 A4⋊C8 C8×A4 C4×S4 A4⋊C4 C2×S4 S4 C22×C8 C22×C4 C23 C22 C8 C4 C2 C1 # reps 1 1 1 1 2 2 8 1 1 2 4 2 2 4 8

Matrix representation of C8×S4 in GL3(𝔽73) generated by

 51 0 0 0 51 0 0 0 51
,
 72 0 0 0 72 0 0 0 1
,
 1 0 0 0 72 0 0 0 72
,
 0 0 1 1 0 0 0 1 0
,
 1 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(73))| [51,0,0,0,51,0,0,0,51],[72,0,0,0,72,0,0,0,1],[1,0,0,0,72,0,0,0,72],[0,1,0,0,0,1,1,0,0],[1,0,0,0,0,1,0,1,0] >;

C8×S4 in GAP, Magma, Sage, TeX

C_8\times S_4
% in TeX

G:=Group("C8xS4");
// GroupNames label

G:=SmallGroup(192,958);
// by ID

G=gap.SmallGroup(192,958);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,36,58,1124,4037,285,2358,475]);
// Polycyclic

G:=Group<a,b,c,d,e|a^8=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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