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## G = Dic3×D12order 288 = 25·32

### Direct product of Dic3 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — Dic3×D12
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×S3×Dic3 — Dic3×D12
 Lower central C32 — C3×C6 — Dic3×D12
 Upper central C1 — C22 — C2×C4

Generators and relations for Dic3×D12
G = < a,b,c,d | a6=c12=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 682 in 201 conjugacy classes, 72 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×8], S3 [×4], C6 [×6], C6 [×7], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], C32, Dic3 [×2], Dic3 [×7], C12 [×4], C12 [×5], D6 [×4], D6 [×4], C2×C6 [×2], C2×C6 [×9], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C3×S3 [×4], C3×C6 [×3], C4×S3 [×4], D12 [×4], C2×Dic3 [×2], C2×Dic3 [×10], C2×C12 [×2], C2×C12 [×3], C3×D4 [×4], C22×S3 [×2], C22×C6 [×2], C4×D4, C3×Dic3 [×2], C3×Dic3, C3⋊Dic3 [×2], C3×C12 [×2], S3×C6 [×4], S3×C6 [×4], C62, C4×Dic3, C4⋊Dic3 [×3], D6⋊C4 [×2], C6.D4 [×2], C4×C12, S3×C2×C4 [×2], C2×D12, C22×Dic3 [×2], C6×D4, S3×Dic3 [×4], C3×D12 [×4], C6×Dic3 [×2], C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6 [×2], C4×D12, D4×Dic3, D6⋊Dic3 [×2], Dic3×C12, C12⋊Dic3, C2×S3×Dic3 [×2], C6×D12, Dic3×D12
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], D4 [×2], C23, Dic3 [×4], D6 [×6], C22×C4, C2×D4, C4○D4, C4×S3 [×2], D12 [×2], C2×Dic3 [×6], C22×S3 [×2], C4×D4, S32, S3×C2×C4, C2×D12, C4○D12, S3×D4, D42S3, C22×Dic3, S3×Dic3 [×2], C2×S32, C4×D12, D4×Dic3, D125S3, S3×D12, C2×S3×Dic3, Dic3×D12

Smallest permutation representation of Dic3×D12
On 96 points
Generators in S96
(1 25 9 33 5 29)(2 26 10 34 6 30)(3 27 11 35 7 31)(4 28 12 36 8 32)(13 94 21 90 17 86)(14 95 22 91 18 87)(15 96 23 92 19 88)(16 85 24 93 20 89)(37 69 41 61 45 65)(38 70 42 62 46 66)(39 71 43 63 47 67)(40 72 44 64 48 68)(49 82 53 74 57 78)(50 83 54 75 58 79)(51 84 55 76 59 80)(52 73 56 77 60 81)
(1 65 33 41)(2 66 34 42)(3 67 35 43)(4 68 36 44)(5 69 25 45)(6 70 26 46)(7 71 27 47)(8 72 28 48)(9 61 29 37)(10 62 30 38)(11 63 31 39)(12 64 32 40)(13 77 90 52)(14 78 91 53)(15 79 92 54)(16 80 93 55)(17 81 94 56)(18 82 95 57)(19 83 96 58)(20 84 85 59)(21 73 86 60)(22 74 87 49)(23 75 88 50)(24 76 89 51)
(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 58)(2 57)(3 56)(4 55)(5 54)(6 53)(7 52)(8 51)(9 50)(10 49)(11 60)(12 59)(13 71)(14 70)(15 69)(16 68)(17 67)(18 66)(19 65)(20 64)(21 63)(22 62)(23 61)(24 72)(25 79)(26 78)(27 77)(28 76)(29 75)(30 74)(31 73)(32 84)(33 83)(34 82)(35 81)(36 80)(37 88)(38 87)(39 86)(40 85)(41 96)(42 95)(43 94)(44 93)(45 92)(46 91)(47 90)(48 89)

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

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

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

54 conjugacy classes

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

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 Dic3 D6 D6 D6 C4○D4 D12 C4×S3 C4○D12 S32 S3×D4 D4⋊2S3 S3×Dic3 C2×S32 D12⋊5S3 S3×D12 kernel Dic3×D12 D6⋊Dic3 Dic3×C12 C12⋊Dic3 C2×S3×Dic3 C6×D12 C3×D12 C4×Dic3 C2×D12 C3×Dic3 D12 C2×Dic3 C2×C12 C22×S3 C3×C6 Dic3 C12 C6 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 2 1 1 2 1 8 1 1 2 4 2 2 2 2 4 4 4 1 1 1 2 1 2 2

Matrix representation of Dic3×D12 in GL6(𝔽13)

 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 0 12 0 0 0 0 1 12
,
 5 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
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 1 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 2 7 0 0 0 0 7 11 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

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,7,0,0,0,0,7,11,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic3×D12 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times D_{12}
% in TeX

G:=Group("Dic3xD12");
// GroupNames label

G:=SmallGroup(288,540);
// by ID

G=gap.SmallGroup(288,540);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,422,100,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^12=d^2=1,b^2=a^3,b*a*b^-1=a^-1,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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