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## G = S3×C4○D8order 192 = 26·3

### Direct product of S3 and C4○D8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — S3×C4○D8
 Chief series C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — S3×C4○D4 — S3×C4○D8
 Lower central C3 — C6 — C12 — S3×C4○D8
 Upper central C1 — C4 — C2×C4 — C4○D8

Generators and relations for S3×C4○D8
G = < a,b,c,d,e | a3=b2=c4=e2=1, d4=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d3 >

Subgroups: 712 in 266 conjugacy classes, 101 normal (51 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, C2×C8, D8, D8, SD16, SD16, Q16, Q16, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×S3, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C4○D8, C2×C4○D4, S3×C8, C24⋊C2, D24, Dic12, C2×C3⋊C8, D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C2×C24, C3×D8, C3×SD16, C3×Q16, S3×C2×C4, S3×C2×C4, C4○D12, C4○D12, S3×D4, S3×D4, D42S3, D42S3, S3×Q8, Q83S3, C3×C4○D4, C2×C4○D8, S3×C2×C8, C4○D24, S3×D8, D83S3, S3×SD16, Q8.7D6, S3×Q16, D24⋊C2, Q8.13D6, C3×C4○D8, S3×C4○D4, S3×C4○D8
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C4○D8, C22×D4, S3×D4, S3×C23, C2×C4○D8, C2×S3×D4, S3×C4○D8

Smallest permutation representation of S3×C4○D8
On 48 points
Generators in S48
(1 40 27)(2 33 28)(3 34 29)(4 35 30)(5 36 31)(6 37 32)(7 38 25)(8 39 26)(9 24 45)(10 17 46)(11 18 47)(12 19 48)(13 20 41)(14 21 42)(15 22 43)(16 23 44)
(9 45)(10 46)(11 47)(12 48)(13 41)(14 42)(15 43)(16 44)(25 38)(26 39)(27 40)(28 33)(29 34)(30 35)(31 36)(32 37)
(1 19 5 23)(2 20 6 24)(3 21 7 17)(4 22 8 18)(9 28 13 32)(10 29 14 25)(11 30 15 26)(12 31 16 27)(33 41 37 45)(34 42 38 46)(35 43 39 47)(36 44 40 48)
(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)
(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(17 21)(18 20)(22 24)(25 29)(26 28)(30 32)(33 39)(34 38)(35 37)(41 47)(42 46)(43 45)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 D6 D6 D6 C4○D8 S3×D4 S3×D4 S3×C4○D8 kernel S3×C4○D8 S3×C2×C8 C4○D24 S3×D8 D8⋊3S3 S3×SD16 Q8.7D6 S3×Q16 D24⋊C2 Q8.13D6 C3×C4○D8 S3×C4○D4 C4○D8 C4×S3 C2×Dic3 C22×S3 C2×C8 D8 SD16 Q16 C4○D4 S3 C4 C22 C1 # reps 1 1 1 1 1 2 2 1 1 2 1 2 1 2 1 1 1 1 2 1 2 8 1 1 4

Matrix representation of S3×C4○D8 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 72 72 0 0 1 0
,
 72 0 0 0 0 72 0 0 0 0 1 0 0 0 72 72
,
 27 0 0 0 0 27 0 0 0 0 72 0 0 0 0 72
,
 16 57 0 0 16 16 0 0 0 0 72 0 0 0 0 72
,
 1 0 0 0 0 72 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,72,1,0,0,72,0],[72,0,0,0,0,72,0,0,0,0,1,72,0,0,0,72],[27,0,0,0,0,27,0,0,0,0,72,0,0,0,0,72],[16,16,0,0,57,16,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1] >;

S3×C4○D8 in GAP, Magma, Sage, TeX

S_3\times C_4\circ D_8
% in TeX

G:=Group("S3xC4oD8");
// GroupNames label

G:=SmallGroup(192,1326);
// by ID

G=gap.SmallGroup(192,1326);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,570,185,438,235,102,6278]);
// Polycyclic

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

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