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## G = S3×M5(2)  order 192 = 26·3

### Direct product of S3 and M5(2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — S3×M5(2)
 Chief series C1 — C3 — C6 — C12 — C24 — S3×C8 — S3×C2×C8 — S3×M5(2)
 Lower central C3 — C6 — S3×M5(2)
 Upper central C1 — C8 — M5(2)

Generators and relations for S3×M5(2)
G = < a,b,c,d | a3=b2=c16=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c9 >

Subgroups: 168 in 90 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, C16, C16, C2×C8, C2×C8, C22×C4, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C2×C16, M5(2), M5(2), C22×C8, C3⋊C16, C48, S3×C8, C2×C3⋊C8, C2×C24, S3×C2×C4, C2×M5(2), S3×C16, D6.C8, C12.C8, C3×M5(2), S3×C2×C8, S3×M5(2)
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, D6, C2×C8, C22×C4, C4×S3, C22×S3, M5(2), C22×C8, S3×C8, S3×C2×C4, C2×M5(2), S3×C2×C8, S3×M5(2)

Smallest permutation representation of S3×M5(2)
On 48 points
Generators in S48
(1 41 30)(2 42 31)(3 43 32)(4 44 17)(5 45 18)(6 46 19)(7 47 20)(8 48 21)(9 33 22)(10 34 23)(11 35 24)(12 36 25)(13 37 26)(14 38 27)(15 39 28)(16 40 29)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 36)(18 37)(19 38)(20 39)(21 40)(22 41)(23 42)(24 43)(25 44)(26 45)(27 46)(28 47)(29 48)(30 33)(31 34)(32 35)
(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 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)(34 42)(36 44)(38 46)(40 48)

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

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

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

60 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 12A 12B 12C 16A ··· 16H 16I ··· 16P 24A 24B 24C 24D 24E 24F 48A ··· 48H 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 12 12 12 16 ··· 16 16 ··· 16 24 24 24 24 24 24 48 ··· 48 size 1 1 2 3 3 6 2 1 1 2 3 3 6 2 4 1 1 1 1 2 2 3 3 3 3 6 6 2 2 4 2 ··· 2 6 ··· 6 2 2 2 2 4 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C8 C8 C8 S3 D6 D6 C4×S3 C4×S3 M5(2) S3×C8 S3×C8 S3×M5(2) kernel S3×M5(2) S3×C16 D6.C8 C12.C8 C3×M5(2) S3×C2×C8 S3×C8 C2×C3⋊C8 S3×C2×C4 C4×S3 C2×Dic3 C22×S3 M5(2) C16 C2×C8 C8 C2×C4 S3 C4 C22 C1 # reps 1 2 2 1 1 1 4 2 2 8 4 4 1 2 1 2 2 8 4 4 4

Matrix representation of S3×M5(2) in GL4(𝔽97) generated by

 0 96 0 0 1 96 0 0 0 0 1 0 0 0 0 1
,
 1 96 0 0 0 96 0 0 0 0 96 0 0 0 0 96
,
 64 0 0 0 0 64 0 0 0 0 47 3 0 0 24 50
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 96
G:=sub<GL(4,GF(97))| [0,1,0,0,96,96,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,96,96,0,0,0,0,96,0,0,0,0,96],[64,0,0,0,0,64,0,0,0,0,47,24,0,0,3,50],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,96] >;

S3×M5(2) in GAP, Magma, Sage, TeX

S_3\times M_5(2)
% in TeX

G:=Group("S3xM5(2)");
// GroupNames label

G:=SmallGroup(192,465);
// by ID

G=gap.SmallGroup(192,465);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,58,80,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^16=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^9>;
// generators/relations

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