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## G = S3×C4×C12order 288 = 25·32

### Direct product of C4×C12 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C4×C12
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — S3×C2×C12 — S3×C4×C12
 Lower central C3 — S3×C4×C12
 Upper central C1 — C4×C12

Generators and relations for S3×C4×C12
G = < a,b,c,d | a4=b12=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 402 in 231 conjugacy classes, 138 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×6], C4 [×6], C22, C22 [×6], S3 [×4], C6 [×6], C6 [×7], C2×C4 [×3], C2×C4 [×15], C23, C32, Dic3 [×6], C12 [×12], C12 [×12], D6 [×6], C2×C6 [×2], C2×C6 [×7], C42, C42 [×3], C22×C4 [×3], C3×S3 [×4], C3×C6 [×3], C4×S3 [×12], C2×Dic3 [×3], C2×C12 [×6], C2×C12 [×18], C22×S3, C22×C6, C2×C42, C3×Dic3 [×6], C3×C12 [×6], S3×C6 [×6], C62, C4×Dic3 [×3], C4×C12 [×2], C4×C12 [×4], S3×C2×C4 [×3], C22×C12 [×3], S3×C12 [×12], C6×Dic3 [×3], C6×C12 [×3], S3×C2×C6, S3×C42, C2×C4×C12, Dic3×C12 [×3], C122, S3×C2×C12 [×3], S3×C4×C12
Quotients: C1, C2 [×7], C3, C4 [×12], C22 [×7], S3, C6 [×7], C2×C4 [×18], C23, C12 [×12], D6 [×3], C2×C6 [×7], C42 [×4], C22×C4 [×3], C3×S3, C4×S3 [×6], C2×C12 [×18], C22×S3, C22×C6, C2×C42, S3×C6 [×3], C4×C12 [×4], S3×C2×C4 [×3], C22×C12 [×3], S3×C12 [×6], S3×C2×C6, S3×C42, C2×C4×C12, S3×C2×C12 [×3], S3×C4×C12

Smallest permutation representation of S3×C4×C12
On 96 points
Generators in S96
(1 48 96 62)(2 37 85 63)(3 38 86 64)(4 39 87 65)(5 40 88 66)(6 41 89 67)(7 42 90 68)(8 43 91 69)(9 44 92 70)(10 45 93 71)(11 46 94 72)(12 47 95 61)(13 50 82 34)(14 51 83 35)(15 52 84 36)(16 53 73 25)(17 54 74 26)(18 55 75 27)(19 56 76 28)(20 57 77 29)(21 58 78 30)(22 59 79 31)(23 60 80 32)(24 49 81 33)
(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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(61 65 69)(62 66 70)(63 67 71)(64 68 72)(73 81 77)(74 82 78)(75 83 79)(76 84 80)(85 89 93)(86 90 94)(87 91 95)(88 92 96)
(1 80)(2 81)(3 82)(4 83)(5 84)(6 73)(7 74)(8 75)(9 76)(10 77)(11 78)(12 79)(13 86)(14 87)(15 88)(16 89)(17 90)(18 91)(19 92)(20 93)(21 94)(22 95)(23 96)(24 85)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(33 37)(34 38)(35 39)(36 40)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)(57 71)(58 72)(59 61)(60 62)

G:=sub<Sym(96)| (1,48,96,62)(2,37,85,63)(3,38,86,64)(4,39,87,65)(5,40,88,66)(6,41,89,67)(7,42,90,68)(8,43,91,69)(9,44,92,70)(10,45,93,71)(11,46,94,72)(12,47,95,61)(13,50,82,34)(14,51,83,35)(15,52,84,36)(16,53,73,25)(17,54,74,26)(18,55,75,27)(19,56,76,28)(20,57,77,29)(21,58,78,30)(22,59,79,31)(23,60,80,32)(24,49,81,33), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,65,69)(62,66,70)(63,67,71)(64,68,72)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,89,93)(86,90,94)(87,91,95)(88,92,96), (1,80)(2,81)(3,82)(4,83)(5,84)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,86)(14,87)(15,88)(16,89)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(23,96)(24,85)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,37)(34,38)(35,39)(36,40)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70)(57,71)(58,72)(59,61)(60,62)>;

G:=Group( (1,48,96,62)(2,37,85,63)(3,38,86,64)(4,39,87,65)(5,40,88,66)(6,41,89,67)(7,42,90,68)(8,43,91,69)(9,44,92,70)(10,45,93,71)(11,46,94,72)(12,47,95,61)(13,50,82,34)(14,51,83,35)(15,52,84,36)(16,53,73,25)(17,54,74,26)(18,55,75,27)(19,56,76,28)(20,57,77,29)(21,58,78,30)(22,59,79,31)(23,60,80,32)(24,49,81,33), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,65,69)(62,66,70)(63,67,71)(64,68,72)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,89,93)(86,90,94)(87,91,95)(88,92,96), (1,80)(2,81)(3,82)(4,83)(5,84)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,86)(14,87)(15,88)(16,89)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(23,96)(24,85)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,37)(34,38)(35,39)(36,40)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70)(57,71)(58,72)(59,61)(60,62) );

G=PermutationGroup([(1,48,96,62),(2,37,85,63),(3,38,86,64),(4,39,87,65),(5,40,88,66),(6,41,89,67),(7,42,90,68),(8,43,91,69),(9,44,92,70),(10,45,93,71),(11,46,94,72),(12,47,95,61),(13,50,82,34),(14,51,83,35),(15,52,84,36),(16,53,73,25),(17,54,74,26),(18,55,75,27),(19,56,76,28),(20,57,77,29),(21,58,78,30),(22,59,79,31),(23,60,80,32),(24,49,81,33)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,41,45),(38,42,46),(39,43,47),(40,44,48),(49,57,53),(50,58,54),(51,59,55),(52,60,56),(61,65,69),(62,66,70),(63,67,71),(64,68,72),(73,81,77),(74,82,78),(75,83,79),(76,84,80),(85,89,93),(86,90,94),(87,91,95),(88,92,96)], [(1,80),(2,81),(3,82),(4,83),(5,84),(6,73),(7,74),(8,75),(9,76),(10,77),(11,78),(12,79),(13,86),(14,87),(15,88),(16,89),(17,90),(18,91),(19,92),(20,93),(21,94),(22,95),(23,96),(24,85),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48),(33,37),(34,38),(35,39),(36,40),(49,63),(50,64),(51,65),(52,66),(53,67),(54,68),(55,69),(56,70),(57,71),(58,72),(59,61),(60,62)])

144 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A ··· 4L 4M ··· 4X 6A ··· 6F 6G ··· 6O 6P ··· 6W 12A ··· 12X 12Y ··· 12BH 12BI ··· 12CF order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 ··· 4 4 ··· 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 3 3 3 3 1 1 2 2 2 1 ··· 1 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 D6 C3×S3 C4×S3 S3×C6 S3×C12 kernel S3×C4×C12 Dic3×C12 C122 S3×C2×C12 S3×C42 S3×C12 C4×Dic3 C4×C12 S3×C2×C4 C4×S3 C4×C12 C2×C12 C42 C12 C2×C4 C4 # reps 1 3 1 3 2 24 6 2 6 48 1 3 2 12 6 24

Matrix representation of S3×C4×C12 in GL3(𝔽13) generated by

 1 0 0 0 8 0 0 0 8
,
 7 0 0 0 7 0 0 0 7
,
 1 0 0 0 9 0 0 6 3
,
 12 0 0 0 12 1 0 0 1
G:=sub<GL(3,GF(13))| [1,0,0,0,8,0,0,0,8],[7,0,0,0,7,0,0,0,7],[1,0,0,0,9,6,0,0,3],[12,0,0,0,12,0,0,1,1] >;

S3×C4×C12 in GAP, Magma, Sage, TeX

S_3\times C_4\times C_{12}
% in TeX

G:=Group("S3xC4xC12");
// GroupNames label

G:=SmallGroup(288,642);
// by ID

G=gap.SmallGroup(288,642);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,142,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^12=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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