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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 — C62 — S3×C2×C6 — C2×S3×Dic3 — S3×C4⋊Dic3
 Lower central C32 — C3×C6 — S3×C4⋊Dic3
 Upper central C1 — C22 — C2×C4

Generators and relations for S3×C4⋊Dic3
G = < a,b,c,d,e | a3=b2=c4=d6=1, e2=d3, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 618 in 197 conjugacy classes, 84 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×6], S3 [×4], C6 [×6], C6 [×7], C2×C4, C2×C4 [×13], C23, C32, Dic3 [×2], Dic3 [×8], C12 [×4], C12 [×6], D6 [×6], C2×C6 [×2], C2×C6 [×7], C4⋊C4 [×4], C22×C4 [×3], C3×S3 [×4], C3×C6 [×3], C4×S3 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3 [×2], C2×Dic3 [×10], C2×C12 [×2], C2×C12 [×8], C22×S3, C22×C6, C2×C4⋊C4, C3×Dic3 [×2], C3×Dic3 [×2], C3⋊Dic3 [×2], C3×C12 [×2], S3×C6 [×6], C62, Dic3⋊C4 [×2], C4⋊Dic3, C4⋊Dic3 [×5], C3×C4⋊C4, S3×C2×C4, S3×C2×C4 [×2], C22×Dic3 [×2], C22×C12, S3×Dic3 [×4], S3×C12 [×4], C6×Dic3, C6×Dic3 [×2], C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6, S3×C4⋊C4, C2×C4⋊Dic3, Dic3⋊Dic3 [×2], C3×C4⋊Dic3, C12⋊Dic3, C2×S3×Dic3 [×2], S3×C2×C12, S3×C4⋊Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], D4 [×2], Q8 [×2], C23, Dic3 [×4], D6 [×6], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, Dic6 [×2], C4×S3 [×2], D12 [×2], C2×Dic3 [×6], C22×S3 [×2], C2×C4⋊C4, S32, C4⋊Dic3 [×4], C2×Dic6, S3×C2×C4, C2×D12, S3×D4, S3×Q8, C22×Dic3, S3×Dic3 [×2], C2×S32, S3×C4⋊C4, C2×C4⋊Dic3, S3×Dic6, S3×D12, C2×S3×Dic3, S3×C4⋊Dic3

Smallest permutation representation of S3×C4⋊Dic3
On 96 points
Generators in S96
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(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)(49 53 51)(50 54 52)(55 59 57)(56 60 58)(61 65 63)(62 66 64)(67 71 69)(68 72 70)(73 77 75)(74 78 76)(79 81 83)(80 82 84)(85 87 89)(86 88 90)(91 93 95)(92 94 96)
(1 60)(2 55)(3 56)(4 57)(5 58)(6 59)(7 51)(8 52)(9 53)(10 54)(11 49)(12 50)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)
(1 27 16 24)(2 28 17 19)(3 29 18 20)(4 30 13 21)(5 25 14 22)(6 26 15 23)(7 81 93 86)(8 82 94 87)(9 83 95 88)(10 84 96 89)(11 79 91 90)(12 80 92 85)(31 43 42 49)(32 44 37 50)(33 45 38 51)(34 46 39 52)(35 47 40 53)(36 48 41 54)(55 76 65 67)(56 77 66 68)(57 78 61 69)(58 73 62 70)(59 74 63 71)(60 75 64 72)
(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 33 4 36)(2 32 5 35)(3 31 6 34)(7 78 10 75)(8 77 11 74)(9 76 12 73)(13 41 16 38)(14 40 17 37)(15 39 18 42)(19 44 22 47)(20 43 23 46)(21 48 24 45)(25 53 28 50)(26 52 29 49)(27 51 30 54)(55 80 58 83)(56 79 59 82)(57 84 60 81)(61 89 64 86)(62 88 65 85)(63 87 66 90)(67 92 70 95)(68 91 71 94)(69 96 72 93)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C ··· 4H 4I 4J 4K 4L 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 12O 12P 12Q 12R order 1 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 12 12 12 12 12 ··· 12 12 12 12 12 12 12 12 12 size 1 1 1 1 3 3 3 3 2 2 4 2 2 6 ··· 6 18 18 18 18 2 ··· 2 4 4 4 6 6 6 6 2 2 2 2 4 ··· 4 6 6 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 2 2 4 4 4 4 4 4 4 type + + + + + + + + + - - + + + - + + + - - + - + image C1 C2 C2 C2 C2 C2 C4 S3 S3 D4 Q8 Dic3 D6 D6 D6 C4×S3 Dic6 D12 S32 S3×D4 S3×Q8 S3×Dic3 C2×S32 S3×Dic6 S3×D12 kernel S3×C4⋊Dic3 Dic3⋊Dic3 C3×C4⋊Dic3 C12⋊Dic3 C2×S3×Dic3 S3×C2×C12 S3×C12 C4⋊Dic3 S3×C2×C4 S3×C6 S3×C6 C4×S3 C2×Dic3 C2×C12 C22×S3 C12 D6 D6 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 2 1 1 2 1 8 1 1 2 2 4 3 2 1 4 4 4 1 1 1 2 1 2 2

Matrix representation of S3×C4⋊Dic3 in GL6(𝔽13)

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

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

S3×C4⋊Dic3 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(288,537);
// by ID

G=gap.SmallGroup(288,537);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,219,100,1356,9414]);
// Polycyclic

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

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