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## G = S3×C4.Dic3order 288 = 25·32

### Direct product of S3 and C4.Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — S3×C4.Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — S3×C3⋊C8 — S3×C4.Dic3
 Lower central C32 — C3×C6 — S3×C4.Dic3
 Upper central C1 — C4 — C2×C4

Generators and relations for S3×C4.Dic3
G = < a,b,c,d,e | a3=b2=c4=1, d6=c2, e2=c2d3, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d5 >

Subgroups: 370 in 146 conjugacy classes, 64 normal (50 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, S3, C6, C6, C8, C2×C4, C2×C4, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C3×S3, C3×S3, C3×C6, C3×C6, C3⋊C8, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C2×M4(2), C3×Dic3, C3×C12, S3×C6, S3×C6, C62, S3×C8, C8⋊S3, C2×C3⋊C8, C4.Dic3, C4.Dic3, C3×M4(2), S3×C2×C4, C22×C12, C3×C3⋊C8, C324C8, S3×C12, C6×Dic3, C6×C12, S3×C2×C6, S3×M4(2), C2×C4.Dic3, S3×C3⋊C8, D6.Dic3, C3×C4.Dic3, C12.58D6, S3×C2×C12, S3×C4.Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, M4(2), C22×C4, C4×S3, C2×Dic3, C22×S3, C2×M4(2), S32, C4.Dic3, S3×C2×C4, C22×Dic3, S3×Dic3, C2×S32, S3×M4(2), C2×C4.Dic3, C2×S3×Dic3, S3×C4.Dic3

Smallest permutation representation of S3×C4.Dic3
On 48 points
Generators in S48
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 36)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 46)(14 47)(15 48)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 16 19 22)(14 17 20 23)(15 18 21 24)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 40 43 46)(38 41 44 47)(39 42 45 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 19 10 16 7 13 4 22)(2 24 11 21 8 18 5 15)(3 17 12 14 9 23 6 20)(25 45 34 42 31 39 28 48)(26 38 35 47 32 44 29 41)(27 43 36 40 33 37 30 46)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 8A 8B 8C 8D 8E 8F 8G 8H 12A ··· 12F 12G ··· 12K 12L 12M 12N 12O 24A 24B 24C 24D order 1 2 2 2 2 2 3 3 3 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 6 6 6 8 8 8 8 8 8 8 8 12 ··· 12 12 ··· 12 12 12 12 12 24 24 24 24 size 1 1 2 3 3 6 2 2 4 1 1 2 3 3 6 2 2 2 2 4 4 4 4 6 6 6 6 6 6 6 6 18 18 18 18 2 ··· 2 4 ··· 4 6 6 6 6 12 12 12 12

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + - + - + - + - + - image C1 C2 C2 C2 C2 C2 C4 C4 C4 S3 S3 D6 Dic3 D6 Dic3 D6 Dic3 M4(2) C4×S3 C4×S3 C4.Dic3 S32 S3×Dic3 C2×S32 S3×Dic3 S3×M4(2) S3×C4.Dic3 kernel S3×C4.Dic3 S3×C3⋊C8 D6.Dic3 C3×C4.Dic3 C12.58D6 S3×C2×C12 S3×C12 C6×Dic3 S3×C2×C6 C4.Dic3 S3×C2×C4 C3⋊C8 C4×S3 C4×S3 C2×Dic3 C2×C12 C22×S3 C3×S3 C12 C2×C6 S3 C2×C4 C4 C4 C22 C3 C1 # reps 1 2 2 1 1 1 4 2 2 1 1 2 2 2 1 2 1 4 2 2 8 1 1 1 1 2 4

Matrix representation of S3×C4.Dic3 in GL6(𝔽73)

 0 1 0 0 0 0 72 72 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
,
 72 0 0 0 0 0 1 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 27 0 0 0 0 0 72 46 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 0 0 0 0 1 72 0 0 0 0 1 0
,
 27 0 0 0 0 0 0 27 0 0 0 0 0 0 27 71 0 0 0 0 59 46 0 0 0 0 0 0 0 27 0 0 0 0 27 0

G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,27,72,0,0,0,0,0,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,1,1,0,0,0,0,72,0],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,27,59,0,0,0,0,71,46,0,0,0,0,0,0,0,27,0,0,0,0,27,0] >;

S3×C4.Dic3 in GAP, Magma, Sage, TeX

S_3\times C_4.{\rm Dic}_3
% in TeX

G:=Group("S3xC4.Dic3");
// GroupNames label

G:=SmallGroup(288,461);
// by ID

G=gap.SmallGroup(288,461);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,219,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^4=1,d^6=c^2,e^2=c^2*d^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,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^5>;
// generators/relations

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