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## G = S3×C8.C4order 192 = 26·3

### Direct product of S3 and C8.C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — S3×C8.C4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — S3×C2×C4 — S3×C2×C8 — S3×C8.C4
 Lower central C3 — C6 — C12 — S3×C8.C4
 Upper central C1 — C4 — C2×C4 — C8.C4

Generators and relations for S3×C8.C4
G = < a,b,c,d | a3=b2=c8=1, d4=c4, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 224 in 106 conjugacy classes, 53 normal (41 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, Dic3, C12, D6, D6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C3⋊C8, C3⋊C8, C24, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C8.C4, C8.C4, C22×C8, C2×M4(2), S3×C8, S3×C8, C8⋊S3, C2×C3⋊C8, C4.Dic3, C2×C24, C3×M4(2), S3×C2×C4, C2×C8.C4, C24.C4, C12.53D4, C3×C8.C4, S3×C2×C8, S3×M4(2), S3×C8.C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4×S3, C22×S3, C8.C4, C2×C4⋊C4, S3×C2×C4, S3×D4, S3×Q8, C2×C8.C4, S3×C4⋊C4, S3×C8.C4

Smallest permutation representation of S3×C8.C4
On 48 points
Generators in S48
(1 13 18)(2 14 19)(3 15 20)(4 16 21)(5 9 22)(6 10 23)(7 11 24)(8 12 17)(25 45 37)(26 46 38)(27 47 39)(28 48 40)(29 41 33)(30 42 34)(31 43 35)(32 44 36)
(1 5)(2 6)(3 7)(4 8)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)(25 29)(26 30)(27 31)(28 32)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 44)
(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 37 38 39 40)(41 42 43 44 45 46 47 48)
(1 27 7 29 5 31 3 25)(2 26 8 28 6 30 4 32)(9 43 15 45 13 47 11 41)(10 42 16 44 14 46 12 48)(17 40 23 34 21 36 19 38)(18 39 24 33 22 35 20 37)

G:=sub<Sym(48)| (1,13,18)(2,14,19)(3,15,20)(4,16,21)(5,9,22)(6,10,23)(7,11,24)(8,12,17)(25,45,37)(26,46,38)(27,47,39)(28,48,40)(29,41,33)(30,42,34)(31,43,35)(32,44,36), (1,5)(2,6)(3,7)(4,8)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17)(25,29)(26,30)(27,31)(28,32)(33,45)(34,46)(35,47)(36,48)(37,41)(38,42)(39,43)(40,44), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,27,7,29,5,31,3,25)(2,26,8,28,6,30,4,32)(9,43,15,45,13,47,11,41)(10,42,16,44,14,46,12,48)(17,40,23,34,21,36,19,38)(18,39,24,33,22,35,20,37)>;

G:=Group( (1,13,18)(2,14,19)(3,15,20)(4,16,21)(5,9,22)(6,10,23)(7,11,24)(8,12,17)(25,45,37)(26,46,38)(27,47,39)(28,48,40)(29,41,33)(30,42,34)(31,43,35)(32,44,36), (1,5)(2,6)(3,7)(4,8)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17)(25,29)(26,30)(27,31)(28,32)(33,45)(34,46)(35,47)(36,48)(37,41)(38,42)(39,43)(40,44), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,27,7,29,5,31,3,25)(2,26,8,28,6,30,4,32)(9,43,15,45,13,47,11,41)(10,42,16,44,14,46,12,48)(17,40,23,34,21,36,19,38)(18,39,24,33,22,35,20,37) );

G=PermutationGroup([[(1,13,18),(2,14,19),(3,15,20),(4,16,21),(5,9,22),(6,10,23),(7,11,24),(8,12,17),(25,45,37),(26,46,38),(27,47,39),(28,48,40),(29,41,33),(30,42,34),(31,43,35),(32,44,36)], [(1,5),(2,6),(3,7),(4,8),(9,18),(10,19),(11,20),(12,21),(13,22),(14,23),(15,24),(16,17),(25,29),(26,30),(27,31),(28,32),(33,45),(34,46),(35,47),(36,48),(37,41),(38,42),(39,43),(40,44)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48)], [(1,27,7,29,5,31,3,25),(2,26,8,28,6,30,4,32),(9,43,15,45,13,47,11,41),(10,42,16,44,14,46,12,48),(17,40,23,34,21,36,19,38),(18,39,24,33,22,35,20,37)]])

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L 8M 8N 8O 8P 12A 12B 12C 24A 24B 24C 24D 24E 24F 24G 24H order 1 2 2 2 2 2 3 4 4 4 4 4 4 6 6 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 12 12 12 24 24 24 24 24 24 24 24 size 1 1 2 3 3 6 2 1 1 2 3 3 6 2 4 2 2 2 2 4 4 4 4 6 6 6 6 12 12 12 12 2 2 4 4 4 4 4 8 8 8 8

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + - - + + + - image C1 C2 C2 C2 C2 C2 C4 S3 D4 Q8 Q8 D6 D6 C4×S3 C8.C4 S3×D4 S3×Q8 S3×C8.C4 kernel S3×C8.C4 C24.C4 C12.53D4 C3×C8.C4 S3×C2×C8 S3×M4(2) S3×C8 C8.C4 C4×S3 C2×Dic3 C22×S3 C2×C8 M4(2) C8 S3 C4 C22 C1 # reps 1 1 2 1 1 2 8 1 2 1 1 1 2 4 8 1 1 4

Matrix representation of S3×C8.C4 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 0 72 0 0 1 72
,
 72 0 0 0 0 72 0 0 0 0 0 1 0 0 1 0
,
 22 0 0 0 0 10 0 0 0 0 72 0 0 0 0 72
,
 0 1 0 0 27 0 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,72],[72,0,0,0,0,72,0,0,0,0,0,1,0,0,1,0],[22,0,0,0,0,10,0,0,0,0,72,0,0,0,0,72],[0,27,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;

S3×C8.C4 in GAP, Magma, Sage, TeX

S_3\times C_8.C_4
% in TeX

G:=Group("S3xC8.C4");
// GroupNames label

G:=SmallGroup(192,451);
// by ID

G=gap.SmallGroup(192,451);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,58,136,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^8=1,d^4=c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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