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## G = Dic3⋊Dic6order 288 = 25·32

### 1st semidirect product of Dic3 and Dic6 acting via Dic6/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — Dic3⋊Dic6
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — Dic3⋊Dic3 — Dic3⋊Dic6
 Lower central C32 — C62 — 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, dbd-1=a3b, dcd-1=c-1 >

Subgroups: 514 in 155 conjugacy classes, 60 normal (22 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×8], C22, C6 [×2], C6 [×4], C6 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C32, Dic3 [×4], Dic3 [×10], C12 [×4], C12 [×8], C2×C6 [×2], C2×C6, C42, C4⋊C4 [×4], C2×Q8 [×2], C3×C6, C3×C6 [×2], Dic6 [×10], C2×Dic3 [×4], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×5], C3×Q8 [×2], C4⋊Q8, C3×Dic3 [×4], C3×Dic3 [×2], C3⋊Dic3 [×2], C3×C12 [×2], C62, C4×Dic3, Dic3⋊C4 [×4], C4⋊Dic3 [×4], C4×C12, C2×Dic6, C2×Dic6 [×3], C6×Q8, C3×Dic6 [×2], C6×Dic3 [×4], C324Q8 [×2], C2×C3⋊Dic3 [×2], C6×C12, C122Q8, Dic3⋊Q8, Dic3⋊Dic3 [×4], Dic3×C12, C6×Dic6, C2×C324Q8, Dic3⋊Dic6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], Q8 [×4], C23, D6 [×6], C2×D4, C2×Q8 [×2], Dic6 [×4], D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C4⋊Q8, S32, C2×Dic6 [×2], C2×D12, S3×Q8 [×2], C2×C3⋊D4, C3⋊D12 [×2], C2×S32, C122Q8, Dic3⋊Q8, S3×Dic6 [×2], C2×C3⋊D12, Dic3⋊Dic6

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

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 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 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 6 6 6 6 12 12 36 36 2 ··· 2 4 4 4 2 2 2 2 4 ··· 4 6 ··· 6 12 12 12 12

48 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + - + + + - + + - + + - image C1 C2 C2 C2 C2 S3 S3 Q8 D4 D6 D6 Dic6 D12 C3⋊D4 S32 S3×Q8 C3⋊D12 C2×S32 S3×Dic6 kernel Dic3⋊Dic6 Dic3⋊Dic3 Dic3×C12 C6×Dic6 C2×C32⋊4Q8 C4×Dic3 C2×Dic6 C3×Dic3 C3×C12 C2×Dic3 C2×C12 Dic3 C12 C12 C2×C4 C6 C4 C22 C2 # reps 1 4 1 1 1 1 1 4 2 4 2 8 4 4 1 2 2 1 4

Matrix representation of Dic3⋊Dic6 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 12 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 8 0 0 0 0 0 11 5 0 0 0 0 0 0 12 0 0 0 0 0 1 1 0 0 0 0 0 0 10 6 0 0 0 0 7 3
,
 8 0 0 0 0 0 11 5 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 1
,
 7 4 0 0 0 0 7 6 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 6 10 0 0 0 0 3 7

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,11,0,0,0,0,0,5,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,10,7,0,0,0,0,6,3],[8,11,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[7,7,0,0,0,0,4,6,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,6,3,0,0,0,0,10,7] >;

Dic3⋊Dic6 in GAP, Magma, Sage, TeX

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

G:=Group("Dic3:Dic6");
// GroupNames label

G:=SmallGroup(288,514);
// by ID

G=gap.SmallGroup(288,514);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,64,422,100,1356,9414]);
// 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,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations

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