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## G = S3×D6⋊C4order 288 = 25·32

### Direct product of S3 and D6⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — S3×D6⋊C4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C22×S32 — S3×D6⋊C4
 Lower central C32 — C3×C6 — S3×D6⋊C4
 Upper central C1 — C22 — C2×C4

Generators and relations for S3×D6⋊C4
G = < a,b,c,d,e | a3=b2=c6=d2=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c-1, ce=ec, ede-1=c3d >

Subgroups: 1242 in 281 conjugacy classes, 74 normal (44 characteristic)
C1, C2 [×3], C2 [×8], C3 [×2], C3, C4 [×4], C22, C22 [×22], S3 [×4], S3 [×8], C6 [×6], C6 [×9], C2×C4, C2×C4 [×7], C23 [×11], C32, Dic3 [×5], C12 [×5], D6 [×8], D6 [×30], C2×C6 [×2], C2×C6 [×11], C22⋊C4 [×4], C22×C4 [×2], C24, C3×S3 [×4], C3×S3 [×2], C3⋊S3 [×2], C3×C6 [×3], C4×S3 [×4], C2×Dic3 [×2], C2×Dic3 [×5], C2×C12 [×2], C2×C12 [×5], C22×S3 [×2], C22×S3 [×19], C22×C6 [×2], C2×C22⋊C4, C3×Dic3 [×2], C3⋊Dic3, C3×C12, S32 [×8], S3×C6 [×8], S3×C6 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, D6⋊C4, D6⋊C4 [×6], C6.D4, C3×C22⋊C4, S3×C2×C4, S3×C2×C4, C22×Dic3, C22×C12, S3×C23 [×2], S3×Dic3 [×2], S3×C12 [×2], C6×Dic3 [×2], C2×C3⋊Dic3, C6×C12, C2×S32 [×4], C2×S32 [×4], S3×C2×C6 [×2], C22×C3⋊S3, S3×C22⋊C4, C2×D6⋊C4, D6⋊Dic3, C6.D12, C3×D6⋊C4, C6.11D12, C2×S3×Dic3, S3×C2×C12, C22×S32, S3×D6⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], D4 [×4], C23, D6 [×6], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×4], D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C2×C22⋊C4, S32, D6⋊C4 [×4], S3×C2×C4 [×2], C2×D12, S3×D4 [×2], C2×C3⋊D4, C2×S32, S3×C22⋊C4, C2×D6⋊C4, C4×S32, S3×D12, S3×C3⋊D4, S3×D6⋊C4

Smallest permutation representation of S3×D6⋊C4
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 23 21)(20 24 22)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 25)(7 32)(8 33)(9 34)(10 35)(11 36)(12 31)(13 41)(14 42)(15 37)(16 38)(17 39)(18 40)(19 47)(20 48)(21 43)(22 44)(23 45)(24 46)
(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 28)(2 27)(3 26)(4 25)(5 30)(6 29)(7 34)(8 33)(9 32)(10 31)(11 36)(12 35)(13 40)(14 39)(15 38)(16 37)(17 42)(18 41)(19 46)(20 45)(21 44)(22 43)(23 48)(24 47)
(1 19 7 13)(2 20 8 14)(3 21 9 15)(4 22 10 16)(5 23 11 17)(6 24 12 18)(25 46 31 40)(26 47 32 41)(27 48 33 42)(28 43 34 37)(29 44 35 38)(30 45 36 39)

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

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

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

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 ··· 6F 6G 6H 6I 6J 6K 6L 6M 6N 6O 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 12O 12P 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 6 6 6 6 6 12 12 12 12 12 ··· 12 12 12 12 12 12 12 size 1 1 1 1 3 3 3 3 6 6 18 18 2 2 4 2 2 6 6 6 6 18 18 2 ··· 2 4 4 4 6 6 6 6 12 12 2 2 2 2 4 ··· 4 6 6 6 6 12 12

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 S3 D4 D6 D6 D6 C4×S3 D12 C3⋊D4 S32 S3×D4 C2×S32 C4×S32 S3×D12 S3×C3⋊D4 kernel S3×D6⋊C4 D6⋊Dic3 C6.D12 C3×D6⋊C4 C6.11D12 C2×S3×Dic3 S3×C2×C12 C22×S32 C2×S32 D6⋊C4 S3×C2×C4 S3×C6 C2×Dic3 C2×C12 C22×S3 D6 D6 D6 C2×C4 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 1 4 2 2 2 8 4 4 1 2 1 2 2 2

Matrix representation of S3×D6⋊C4 in GL6(𝔽13)

 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
,
 12 0 0 0 0 0 1 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 1 1 0 0 0 0 0 0 12 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 5 10 0 0 0 0 0 8 0 0 0 0 0 0 5 0 0 0 0 0 0 5

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

S3×D6⋊C4 in GAP, Magma, Sage, TeX

S_3\times D_6\rtimes C_4
% in TeX

G:=Group("S3xD6:C4");
// GroupNames label

G:=SmallGroup(288,568);
// by ID

G=gap.SmallGroup(288,568);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,58,1356,9414]);
// Polycyclic

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

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