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## G = S3×C4.D4order 192 = 26·3

### Direct product of S3 and C4.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — S3×C4.D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — S3×C2×C4 — C2×S3×D4 — S3×C4.D4
 Lower central C3 — C6 — C2×C6 — S3×C4.D4
 Upper central C1 — C2 — C2×C4 — C4.D4

Generators and relations for S3×C4.D4
G = < a,b,c,d,e | a3=b2=c4=1, d4=c2, e2=c, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, ede-1=c-1d3 >

Subgroups: 656 in 186 conjugacy classes, 53 normal (21 characteristic)
C1, C2, C2 [×8], C3, C4 [×2], C4 [×2], C22, C22 [×20], S3 [×2], S3 [×3], C6, C6 [×3], C8 [×4], C2×C4, C2×C4 [×5], D4 [×8], C23 [×2], C23 [×11], Dic3 [×2], C12 [×2], D6 [×2], D6 [×14], C2×C6, C2×C6 [×4], C2×C8 [×2], M4(2) [×2], M4(2) [×4], C22×C4, C2×D4, C2×D4 [×7], C24 [×2], C3⋊C8 [×2], C24 [×2], C4×S3 [×4], D12 [×2], C2×Dic3, C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3, C22×S3 [×2], C22×S3 [×8], C22×C6 [×2], C4.D4, C4.D4 [×3], C2×M4(2) [×2], C22×D4, S3×C8 [×2], C8⋊S3 [×2], C4.Dic3 [×2], C3×M4(2) [×2], S3×C2×C4, C2×D12, S3×D4 [×4], C2×C3⋊D4 [×2], C6×D4, S3×C23 [×2], C2×C4.D4, C12.46D4 [×2], C12.D4, C3×C4.D4, S3×M4(2) [×2], C2×S3×D4, S3×C4.D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], C22×S3, C4.D4 [×2], C2×C22⋊C4, S3×C2×C4, S3×D4 [×2], C2×C4.D4, S3×C22⋊C4, S3×C4.D4

Permutation representations of S3×C4.D4
On 24 points - transitive group 24T339
Generators in S24
(1 14 22)(2 15 23)(3 16 24)(4 9 17)(5 10 18)(6 11 19)(7 12 20)(8 13 21)
(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)
(1 3 5 7)(2 8 6 4)(9 15 13 11)(10 12 14 16)(17 23 21 19)(18 20 22 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 6 3 4 5 2 7 8)(9 10 15 12 13 14 11 16)(17 18 23 20 21 22 19 24)

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

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

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

G:=TransitiveGroup(24,339);

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 8 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 S3 D4 D6 D6 C4×S3 C4.D4 S3×D4 S3×C4.D4 kernel S3×C4.D4 C12.46D4 C12.D4 C3×C4.D4 S3×M4(2) C2×S3×D4 C2×C3⋊D4 S3×C23 C4.D4 C4×S3 M4(2) C2×D4 C23 S3 C4 C1 # reps 1 2 1 1 2 1 4 4 1 4 2 1 4 2 2 1

Matrix representation of S3×C4.D4 in GL6(𝔽73)

 72 72 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 72 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 3 0 0 0 0 48 1 0 0 0 0 59 0 0 1 0 0 59 42 72 0
,
 27 0 0 0 0 0 0 27 0 0 0 0 0 0 46 42 71 0 0 0 55 27 48 25 0 0 60 35 13 14 0 0 59 4 59 60
,
 46 0 0 0 0 0 0 46 0 0 0 0 0 0 46 42 71 0 0 0 0 1 48 48 0 0 0 0 13 59 0 0 1 0 59 13

G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,48,59,59,0,0,3,1,0,42,0,0,0,0,0,72,0,0,0,0,1,0],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,46,55,60,59,0,0,42,27,35,4,0,0,71,48,13,59,0,0,0,25,14,60],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,1,0,0,42,1,0,0,0,0,71,48,13,59,0,0,0,48,59,13] >;

S3×C4.D4 in GAP, Magma, Sage, TeX

S_3\times C_4.D_4
% in TeX

G:=Group("S3xC4.D4");
// GroupNames label

G:=SmallGroup(192,303);
// by ID

G=gap.SmallGroup(192,303);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,58,570,136,438,6278]);
// Polycyclic

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

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