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## G = S3×C6×D4order 288 = 25·32

### Direct product of C6, S3 and D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — S3×C6×D4
 Chief series C1 — C3 — C6 — C3×C6 — S3×C6 — S3×C2×C6 — S3×C22×C6 — S3×C6×D4
 Lower central C3 — C6 — S3×C6×D4
 Upper central C1 — C2×C6 — C6×D4

Generators and relations for S3×C6×D4
G = < a,b,c,d,e | a6=b3=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1194 in 499 conjugacy classes, 194 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×12], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C22 [×34], S3 [×4], S3 [×4], C6 [×2], C6 [×4], C6 [×23], C2×C4, C2×C4 [×5], D4 [×4], D4 [×12], C23 [×2], C23 [×19], C32, Dic3 [×2], C12 [×4], C12 [×4], D6 [×10], D6 [×20], C2×C6 [×2], C2×C6 [×8], C2×C6 [×47], C22×C4, C2×D4, C2×D4 [×11], C24 [×2], C3×S3 [×4], C3×S3 [×4], C3×C6, C3×C6 [×2], C3×C6 [×4], C4×S3 [×4], D12 [×4], C2×Dic3, C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×6], C3×D4 [×8], C3×D4 [×16], C22×S3, C22×S3 [×10], C22×S3 [×8], C22×C6 [×4], C22×C6 [×21], C22×D4, C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×10], S3×C6 [×20], C62, C62 [×4], C62 [×4], S3×C2×C4, C2×D12, S3×D4 [×8], C2×C3⋊D4 [×2], C22×C12, C6×D4 [×2], C6×D4 [×12], S3×C23 [×2], C23×C6 [×2], S3×C12 [×4], C3×D12 [×4], C6×Dic3, C3×C3⋊D4 [×8], C6×C12, D4×C32 [×4], S3×C2×C6, S3×C2×C6 [×10], S3×C2×C6 [×8], C2×C62 [×2], C2×S3×D4, D4×C2×C6, S3×C2×C12, C6×D12, C3×S3×D4 [×8], C6×C3⋊D4 [×2], D4×C3×C6, S3×C22×C6 [×2], S3×C6×D4
Quotients: C1, C2 [×15], C3, C22 [×35], S3, C6 [×15], D4 [×4], C23 [×15], D6 [×7], C2×C6 [×35], C2×D4 [×6], C24, C3×S3, C3×D4 [×4], C22×S3 [×7], C22×C6 [×15], C22×D4, S3×C6 [×7], S3×D4 [×2], C6×D4 [×6], S3×C23, C23×C6, S3×C2×C6 [×7], C2×S3×D4, D4×C2×C6, C3×S3×D4 [×2], S3×C22×C6, S3×C6×D4

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

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A ··· 6F 6G ··· 6W 6X ··· 6AE 6AF ··· 6AQ 6AR ··· 6AY 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 6 ··· 6 6 ··· 6 12 12 12 12 12 ··· 12 12 12 12 12 size 1 1 1 1 2 2 2 2 3 3 3 3 6 6 6 6 1 1 2 2 2 2 2 6 6 1 ··· 1 2 ··· 2 3 ··· 3 4 ··· 4 6 ··· 6 2 2 2 2 4 ··· 4 6 6 6 6

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 S3 D4 D6 D6 D6 C3×S3 C3×D4 S3×C6 S3×C6 S3×C6 S3×D4 C3×S3×D4 kernel S3×C6×D4 S3×C2×C12 C6×D12 C3×S3×D4 C6×C3⋊D4 D4×C3×C6 S3×C22×C6 C2×S3×D4 S3×C2×C4 C2×D12 S3×D4 C2×C3⋊D4 C6×D4 S3×C23 C6×D4 S3×C6 C2×C12 C3×D4 C22×C6 C2×D4 D6 C2×C4 D4 C23 C6 C2 # reps 1 1 1 8 2 1 2 2 2 2 16 4 2 4 1 4 1 4 2 2 8 2 8 4 2 4

Matrix representation of S3×C6×D4 in GL4(𝔽13) generated by

 4 0 0 0 0 4 0 0 0 0 12 0 0 0 0 12
,
 3 0 0 0 9 9 0 0 0 0 1 0 0 0 0 1
,
 1 5 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 1 0 0 0 0 1 0 0 0 0 12 2 0 0 12 1
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 12
G:=sub<GL(4,GF(13))| [4,0,0,0,0,4,0,0,0,0,12,0,0,0,0,12],[3,9,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,5,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,12,0,0,2,1],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,12] >;

S3×C6×D4 in GAP, Magma, Sage, TeX

S_3\times C_6\times D_4
% in TeX

G:=Group("S3xC6xD4");
// GroupNames label

G:=SmallGroup(288,992);
// by ID

G=gap.SmallGroup(288,992);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,409,9414]);
// Polycyclic

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

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