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## G = S3×2- 1+4order 192 = 26·3

### Direct product of S3 and 2- 1+4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — S3×2- 1+4
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C2×C4 — C2×S3×Q8 — S3×2- 1+4
 Lower central C3 — C6 — S3×2- 1+4
 Upper central C1 — C2 — 2- 1+4

Generators and relations for S3×2- 1+4
G = < a,b,c,d,e,f | a3=b2=c4=d2=1, e2=f2=c2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef-1=c2e >

Subgroups: 1512 in 794 conjugacy classes, 445 normal (8 characteristic)
C1, C2, C2 [×12], C3, C4 [×10], C4 [×10], C22 [×5], C22 [×16], S3 [×2], S3 [×5], C6, C6 [×5], C2×C4 [×15], C2×C4 [×55], D4 [×10], D4 [×30], Q8 [×10], Q8 [×30], C23 [×5], Dic3 [×10], C12 [×10], D6, D6 [×5], D6 [×10], C2×C6 [×5], C22×C4 [×15], C2×D4 [×10], C2×Q8 [×5], C2×Q8 [×45], C4○D4 [×10], C4○D4 [×70], Dic6 [×30], C4×S3 [×40], D12 [×10], C2×Dic3 [×15], C3⋊D4 [×20], C2×C12 [×15], C3×D4 [×10], C3×Q8 [×10], C22×S3 [×5], C22×Q8 [×5], C2×C4○D4 [×10], 2- 1+4, 2- 1+4 [×15], C2×Dic6 [×15], S3×C2×C4 [×15], C4○D12 [×30], S3×D4 [×10], D42S3 [×30], S3×Q8 [×30], Q83S3 [×10], C6×Q8 [×5], C3×C4○D4 [×10], C2×2- 1+4, C2×S3×Q8 [×5], Q8.15D6 [×5], S3×C4○D4 [×10], Q8○D12 [×10], C3×2- 1+4, S3×2- 1+4
Quotients: C1, C2 [×31], C22 [×155], S3, C23 [×155], D6 [×15], C24 [×31], C22×S3 [×35], 2- 1+4 [×2], C25, S3×C23 [×15], C2×2- 1+4, S3×C24, S3×2- 1+4

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

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

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

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

51 conjugacy classes

 class 1 2A 2B ··· 2F 2G 2H 2I ··· 2M 3 4A ··· 4J 4K ··· 4T 6A 6B ··· 6F 12A ··· 12J order 1 2 2 ··· 2 2 2 2 ··· 2 3 4 ··· 4 4 ··· 4 6 6 ··· 6 12 ··· 12 size 1 1 2 ··· 2 3 3 6 ··· 6 2 2 ··· 2 6 ··· 6 2 4 ··· 4 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 8 type + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 S3 D6 D6 2- 1+4 S3×2- 1+4 kernel S3×2- 1+4 C2×S3×Q8 Q8.15D6 S3×C4○D4 Q8○D12 C3×2- 1+4 2- 1+4 C2×Q8 C4○D4 S3 C1 # reps 1 5 5 10 10 1 1 5 10 2 1

Matrix representation of S3×2- 1+4 in GL6(𝔽13)

 0 12 0 0 0 0 1 12 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
,
 0 12 0 0 0 0 12 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 0 1 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 5 0 8 0 0 0 1 5 0 8
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 0 10 0 0 0 8 1 0 11 0 0 8 0 5 0 0 0 10 0 1 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 8 0 0 0 0 0 1 5 0 0 0 0 0 0 8 0 0 0 2 0 8 5
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 2 0 0 0 0 0 5 0 0 0 0 12 9 8 10 0 0 12 8 0 5

G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,5,1,0,0,0,5,0,5,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,8,8,10,0,0,0,1,0,0,0,0,10,0,5,1,0,0,0,11,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,1,0,2,0,0,0,5,0,0,0,0,0,0,8,8,0,0,0,0,0,5],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,12,12,0,0,2,5,9,8,0,0,0,0,8,0,0,0,0,0,10,5] >;

S3×2- 1+4 in GAP, Magma, Sage, TeX

S_3\times 2_-^{1+4}
% in TeX

G:=Group("S3xES-(2,2)");
// GroupNames label

G:=SmallGroup(192,1526);
// by ID

G=gap.SmallGroup(192,1526);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,297,136,851,6278]);
// Polycyclic

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

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