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## G = S3×C6.D4order 288 = 25·32

### Direct product of S3 and C6.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — S3×C6.D4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×S3×Dic3 — S3×C6.D4
 Lower central C32 — C3×C6 — S3×C6.D4
 Upper central C1 — C22 — C23

Generators and relations for S3×C6.D4
G = < a,b,c,d,e | a3=b2=c6=d4=1, e2=c3, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece-1=c-1, ede-1=c3d-1 >

Subgroups: 922 in 281 conjugacy classes, 84 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×8], C3 [×2], C3, C4 [×4], C22, C22 [×2], C22 [×20], S3 [×4], S3 [×2], C6 [×2], C6 [×4], C6 [×15], C2×C4 [×8], C23, C23 [×10], C32, Dic3 [×8], C12 [×2], D6 [×8], D6 [×10], C2×C6 [×2], C2×C6 [×4], C2×C6 [×27], C22⋊C4 [×4], C22×C4 [×2], C24, C3×S3 [×4], C3×S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×2], C4×S3 [×4], C2×Dic3 [×2], C2×Dic3 [×10], C2×C12 [×2], C22×S3 [×2], C22×S3 [×4], C22×S3 [×4], C22×C6 [×2], C22×C6 [×11], C2×C22⋊C4, C3×Dic3 [×2], C3⋊Dic3 [×2], S3×C6 [×8], S3×C6 [×10], C62, C62 [×2], C62 [×2], D6⋊C4 [×2], C6.D4, C6.D4 [×5], C3×C22⋊C4, S3×C2×C4 [×2], C22×Dic3 [×2], S3×C23, C23×C6, S3×Dic3 [×4], C6×Dic3 [×2], C2×C3⋊Dic3 [×2], S3×C2×C6 [×2], S3×C2×C6 [×4], S3×C2×C6 [×4], C2×C62, S3×C22⋊C4, C2×C6.D4, D6⋊Dic3 [×2], C3×C6.D4, C625C4, C2×S3×Dic3 [×2], S3×C22×C6, S3×C6.D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], D4 [×4], C23, Dic3 [×4], D6 [×6], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], C2×Dic3 [×6], C3⋊D4 [×4], C22×S3 [×2], C2×C22⋊C4, S32, C6.D4 [×4], S3×C2×C4, S3×D4 [×2], C22×Dic3, C2×C3⋊D4 [×2], S3×Dic3 [×2], C2×S32, S3×C22⋊C4, C2×C6.D4, C2×S3×Dic3, S3×C3⋊D4 [×2], S3×C6.D4

Smallest permutation representation of S3×C6.D4
On 48 points
Generators in S48
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 47 45)(44 48 46)
(1 39)(2 40)(3 41)(4 42)(5 37)(6 38)(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(13 36)(14 31)(15 32)(16 33)(17 34)(18 35)(19 43)(20 44)(21 45)(22 46)(23 47)(24 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)
(1 23 14 30)(2 22 15 29)(3 21 16 28)(4 20 17 27)(5 19 18 26)(6 24 13 25)(7 42 44 34)(8 41 45 33)(9 40 46 32)(10 39 47 31)(11 38 48 36)(12 37 43 35)
(1 20 4 23)(2 19 5 22)(3 24 6 21)(7 34 10 31)(8 33 11 36)(9 32 12 35)(13 28 16 25)(14 27 17 30)(15 26 18 29)(37 46 40 43)(38 45 41 48)(39 44 42 47)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6J 6K ··· 6S 6T ··· 6AA 12A 12B 12C 12D order 1 2 2 2 2 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 12 12 12 size 1 1 1 1 2 2 3 3 3 3 6 6 2 2 4 6 6 6 6 18 18 18 18 2 ··· 2 4 ··· 4 6 ··· 6 12 12 12 12

54 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + - + + + + - + image C1 C2 C2 C2 C2 C2 C4 S3 S3 D4 D6 Dic3 D6 D6 C3⋊D4 C4×S3 S32 S3×D4 S3×Dic3 C2×S32 S3×C3⋊D4 kernel S3×C6.D4 D6⋊Dic3 C3×C6.D4 C62⋊5C4 C2×S3×Dic3 S3×C22×C6 S3×C2×C6 C6.D4 S3×C23 S3×C6 C2×Dic3 C22×S3 C22×S3 C22×C6 D6 C2×C6 C23 C6 C22 C22 C2 # reps 1 2 1 1 2 1 8 1 1 4 2 4 2 2 8 4 1 2 2 1 4

Matrix representation of S3×C6.D4 in GL8(𝔽13)

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

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

S3×C6.D4 in GAP, Magma, Sage, TeX

S_3\times C_6.D_4
% in TeX

G:=Group("S3xC6.D4");
// GroupNames label

G:=SmallGroup(288,616);
// by ID

G=gap.SmallGroup(288,616);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,219,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^6=d^4=1,e^2=c^3,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=e*c*e^-1=c^-1,e*d*e^-1=c^3*d^-1>;
// generators/relations

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