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## G = C2×C4×S4order 192 = 26·3

### Direct product of C2×C4 and S4

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C4×S4
G = < a,b,c,d,e,f | a2=b4=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: 766 in 233 conjugacy classes, 43 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×2], C4 [×10], C22 [×2], C22 [×26], S3 [×4], C6 [×3], C2×C4, C2×C4 [×30], D4 [×16], C23, C23 [×2], C23 [×14], Dic3 [×2], C12 [×2], A4, D6 [×6], C2×C6, C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×15], C2×D4 [×12], C24, C24, C4×S3 [×4], C2×Dic3, C2×C12, S4 [×4], C2×A4, C2×A4 [×2], C22×S3, C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C23×C4, C23×C4, C22×D4, A4⋊C4 [×2], C4×A4 [×2], S3×C2×C4, C2×S4 [×6], C22×A4, C2×C4×D4, C4×S4 [×4], C2×A4⋊C4, C2×C4×A4, C22×S4, C2×C4×S4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D6 [×3], C22×C4, C4×S3 [×2], S4, C22×S3, S3×C2×C4, C2×S4 [×3], C4×S4 [×2], C22×S4, C2×C4×S4

Permutation representations of C2×C4×S4
On 24 points - transitive group 24T397
Generators in S24
(1 8)(2 5)(3 6)(4 7)(9 16)(10 13)(11 14)(12 15)(17 24)(18 21)(19 22)(20 23)
(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 6)(2 7)(3 8)(4 5)(9 14)(10 15)(11 16)(12 13)
(9 14)(10 15)(11 16)(12 13)(17 22)(18 23)(19 24)(20 21)
(1 9 23)(2 10 24)(3 11 21)(4 12 22)(5 13 17)(6 14 18)(7 15 19)(8 16 20)
(1 6)(2 7)(3 8)(4 5)(9 18)(10 19)(11 20)(12 17)(13 22)(14 23)(15 24)(16 21)

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

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

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

G:=TransitiveGroup(24,397);

On 24 points - transitive group 24T418
Generators in S24
(1 23)(2 24)(3 21)(4 22)(5 15)(6 16)(7 13)(8 14)(9 18)(10 19)(11 20)(12 17)
(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 3)(2 4)(9 11)(10 12)(17 19)(18 20)(21 23)(22 24)
(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)
(1 11 13)(2 12 14)(3 9 15)(4 10 16)(5 21 18)(6 22 19)(7 23 20)(8 24 17)
(1 21)(2 22)(3 23)(4 24)(5 11)(6 12)(7 9)(8 10)(13 18)(14 19)(15 20)(16 17)

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

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

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

G:=TransitiveGroup(24,418);

40 conjugacy classes

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

40 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 3 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 D6 D6 C4×S3 S4 C2×S4 C2×S4 C4×S4 kernel C2×C4×S4 C4×S4 C2×A4⋊C4 C2×C4×A4 C22×S4 C2×S4 C23×C4 C22×C4 C24 C23 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 8 1 2 1 4 2 4 2 8

Matrix representation of C2×C4×S4 in GL5(𝔽13)

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

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

C2×C4×S4 in GAP, Magma, Sage, TeX

C_2\times C_4\times S_4
% in TeX

G:=Group("C2xC4xS4");
// GroupNames label

G:=SmallGroup(192,1469);
// by ID

G=gap.SmallGroup(192,1469);
# by ID

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

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