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## G = Dic3×Dic6order 288 = 25·32

### Direct product of Dic3 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — Dic3×Dic6
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — Dic32 — Dic3×Dic6
 Lower central C32 — C3×C6 — Dic3×Dic6
 Upper central C1 — C22 — C2×C4

Generators and relations for Dic3×Dic6
G = < a,b,c,d | a6=c12=1, b2=a3, d2=c6, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 426 in 153 conjugacy classes, 72 normal (34 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×9], C22, C6 [×6], C6 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C32, Dic3 [×6], Dic3 [×7], C12 [×4], C12 [×9], C2×C6 [×2], C2×C6, C42 [×3], C4⋊C4 [×3], C2×Q8, C3×C6 [×3], Dic6 [×4], C2×Dic3 [×2], C2×Dic3 [×2], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×5], C3×Q8 [×4], C4×Q8, C3×Dic3 [×6], C3×Dic3, C3⋊Dic3 [×2], C3×C12 [×2], C62, C4×Dic3, C4×Dic3 [×4], Dic3⋊C4 [×2], C4⋊Dic3 [×5], C4×C12, C2×Dic6, C6×Q8, C3×Dic6 [×4], C6×Dic3 [×2], C6×Dic3 [×2], C2×C3⋊Dic3 [×2], C6×C12, C4×Dic6, Q8×Dic3, Dic32 [×2], Dic3⋊Dic3 [×2], Dic3×C12, C12⋊Dic3, C6×Dic6, Dic3×Dic6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], Q8 [×2], C23, Dic3 [×4], D6 [×6], C22×C4, C2×Q8, C4○D4, Dic6 [×2], C4×S3 [×2], C2×Dic3 [×6], C22×S3 [×2], C4×Q8, S32, C2×Dic6, S3×C2×C4, C4○D12, S3×Q8, Q83S3, C22×Dic3, S3×Dic3 [×2], C2×S32, C4×Dic6, Q8×Dic3, S3×Dic6, D6.6D6, C2×S3×Dic3, Dic3×Dic6

Smallest permutation representation of Dic3×Dic6
On 96 points
Generators in S96
(1 55 5 59 9 51)(2 56 6 60 10 52)(3 57 7 49 11 53)(4 58 8 50 12 54)(13 35 21 31 17 27)(14 36 22 32 18 28)(15 25 23 33 19 29)(16 26 24 34 20 30)(37 96 41 88 45 92)(38 85 42 89 46 93)(39 86 43 90 47 94)(40 87 44 91 48 95)(61 79 69 75 65 83)(62 80 70 76 66 84)(63 81 71 77 67 73)(64 82 72 78 68 74)
(1 36 59 18)(2 25 60 19)(3 26 49 20)(4 27 50 21)(5 28 51 22)(6 29 52 23)(7 30 53 24)(8 31 54 13)(9 32 55 14)(10 33 56 15)(11 34 57 16)(12 35 58 17)(37 64 88 78)(38 65 89 79)(39 66 90 80)(40 67 91 81)(41 68 92 82)(42 69 93 83)(43 70 94 84)(44 71 95 73)(45 72 96 74)(46 61 85 75)(47 62 86 76)(48 63 87 77)
(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 65 7 71)(2 64 8 70)(3 63 9 69)(4 62 10 68)(5 61 11 67)(6 72 12 66)(13 43 19 37)(14 42 20 48)(15 41 21 47)(16 40 22 46)(17 39 23 45)(18 38 24 44)(25 88 31 94)(26 87 32 93)(27 86 33 92)(28 85 34 91)(29 96 35 90)(30 95 36 89)(49 77 55 83)(50 76 56 82)(51 75 57 81)(52 74 58 80)(53 73 59 79)(54 84 60 78)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G ··· 4L 4M 4N 4O 4P 6A ··· 6F 6G 6H 6I 12A 12B 12C 12D 12E ··· 12J 12K ··· 12R 12S 12T 12U 12V order 1 2 2 2 3 3 3 4 4 4 4 4 4 4 ··· 4 4 4 4 4 6 ··· 6 6 6 6 12 12 12 12 12 ··· 12 12 ··· 12 12 12 12 12 size 1 1 1 1 2 2 4 2 2 3 3 3 3 6 ··· 6 18 18 18 18 2 ··· 2 4 4 4 2 2 2 2 4 ··· 4 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 4 4 4 4 4 4 4 type + + + + + + + + - - + + - + - + - + - + image C1 C2 C2 C2 C2 C2 C4 S3 S3 Q8 Dic3 D6 D6 C4○D4 Dic6 C4×S3 C4○D12 S32 S3×Q8 Q8⋊3S3 S3×Dic3 C2×S32 S3×Dic6 D6.6D6 kernel Dic3×Dic6 Dic32 Dic3⋊Dic3 Dic3×C12 C12⋊Dic3 C6×Dic6 C3×Dic6 C4×Dic3 C2×Dic6 C3×Dic3 Dic6 C2×Dic3 C2×C12 C3×C6 Dic3 C12 C6 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 2 2 1 1 1 8 1 1 2 4 4 2 2 4 4 4 1 1 1 2 1 2 2

Matrix representation of Dic3×Dic6 in GL6(𝔽13)

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

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

Dic3×Dic6 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times {\rm Dic}_6
% in TeX

G:=Group("Dic3xDic6");
// GroupNames label

G:=SmallGroup(288,490);
// by ID

G=gap.SmallGroup(288,490);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,422,100,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^12=1,b^2=a^3,d^2=c^6,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^-1=c^-1>;
// generators/relations

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