metabelian, supersoluble, monomial
Aliases: C12.29D12, Dic3⋊3Dic6, C62.36C23, C32⋊5(C4⋊Q8), C6.23(S3×Q8), (C3×Dic3)⋊3Q8, (C3×C12).78D4, C6.77(C2×D12), C3⋊3(C12⋊2Q8), (C2×C12).279D6, (C4×Dic3).3S3, (C2×Dic6).4S3, C2.13(S3×Dic6), C6.11(C2×Dic6), C12.76(C3⋊D4), (C6×C12).95C22, (C2×Dic3).14D6, (C6×Dic6).11C2, (Dic3×C12).7C2, C3⋊1(Dic3⋊Q8), C4.11(C3⋊D12), Dic3⋊Dic3.12C2, (C6×Dic3).142C22, (C2×C4).76S32, C22.93(C2×S32), (C3×C6).87(C2×D4), C6.13(C2×C3⋊D4), (C3×C6).20(C2×Q8), C2.17(C2×C3⋊D12), (C2×C6).55(C22×S3), (C2×C32⋊4Q8).13C2, (C2×C3⋊Dic3).30C22, SmallGroup(288,514)
Series: Derived ►Chief ►Lower central ►Upper central
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, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, Q8, C32, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C2×Q8, C3×C6, C3×C6, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C4⋊Q8, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C62, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×Dic6, C2×Dic6, C6×Q8, C3×Dic6, C6×Dic3, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, C12⋊2Q8, Dic3⋊Q8, Dic3⋊Dic3, Dic3×C12, C6×Dic6, C2×C32⋊4Q8, Dic3⋊Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, Dic6, D12, C3⋊D4, C22×S3, C4⋊Q8, S32, C2×Dic6, C2×D12, S3×Q8, C2×C3⋊D4, C3⋊D12, C2×S32, C12⋊2Q8, Dic3⋊Q8, S3×Dic6, C2×C3⋊D12, 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 78 17 82 21 74)(14 79 18 83 22 75)(15 80 19 84 23 76)(16 81 20 73 24 77)(25 65 33 61 29 69)(26 66 34 62 30 70)(27 67 35 63 31 71)(28 68 36 64 32 72)(37 96 45 92 41 88)(38 85 46 93 42 89)(39 86 47 94 43 90)(40 87 48 95 44 91)
(1 93 59 38)(2 94 60 39)(3 95 49 40)(4 96 50 41)(5 85 51 42)(6 86 52 43)(7 87 53 44)(8 88 54 45)(9 89 55 46)(10 90 56 47)(11 91 57 48)(12 92 58 37)(13 65 82 29)(14 66 83 30)(15 67 84 31)(16 68 73 32)(17 69 74 33)(18 70 75 34)(19 71 76 35)(20 72 77 36)(21 61 78 25)(22 62 79 26)(23 63 80 27)(24 64 81 28)
(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 29 7 35)(2 28 8 34)(3 27 9 33)(4 26 10 32)(5 25 11 31)(6 36 12 30)(13 44 19 38)(14 43 20 37)(15 42 21 48)(16 41 22 47)(17 40 23 46)(18 39 24 45)(49 63 55 69)(50 62 56 68)(51 61 57 67)(52 72 58 66)(53 71 59 65)(54 70 60 64)(73 96 79 90)(74 95 80 89)(75 94 81 88)(76 93 82 87)(77 92 83 86)(78 91 84 85)
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)(13,78,17,82,21,74)(14,79,18,83,22,75)(15,80,19,84,23,76)(16,81,20,73,24,77)(25,65,33,61,29,69)(26,66,34,62,30,70)(27,67,35,63,31,71)(28,68,36,64,32,72)(37,96,45,92,41,88)(38,85,46,93,42,89)(39,86,47,94,43,90)(40,87,48,95,44,91), (1,93,59,38)(2,94,60,39)(3,95,49,40)(4,96,50,41)(5,85,51,42)(6,86,52,43)(7,87,53,44)(8,88,54,45)(9,89,55,46)(10,90,56,47)(11,91,57,48)(12,92,58,37)(13,65,82,29)(14,66,83,30)(15,67,84,31)(16,68,73,32)(17,69,74,33)(18,70,75,34)(19,71,76,35)(20,72,77,36)(21,61,78,25)(22,62,79,26)(23,63,80,27)(24,64,81,28), (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), (1,29,7,35)(2,28,8,34)(3,27,9,33)(4,26,10,32)(5,25,11,31)(6,36,12,30)(13,44,19,38)(14,43,20,37)(15,42,21,48)(16,41,22,47)(17,40,23,46)(18,39,24,45)(49,63,55,69)(50,62,56,68)(51,61,57,67)(52,72,58,66)(53,71,59,65)(54,70,60,64)(73,96,79,90)(74,95,80,89)(75,94,81,88)(76,93,82,87)(77,92,83,86)(78,91,84,85)>;
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)(13,78,17,82,21,74)(14,79,18,83,22,75)(15,80,19,84,23,76)(16,81,20,73,24,77)(25,65,33,61,29,69)(26,66,34,62,30,70)(27,67,35,63,31,71)(28,68,36,64,32,72)(37,96,45,92,41,88)(38,85,46,93,42,89)(39,86,47,94,43,90)(40,87,48,95,44,91), (1,93,59,38)(2,94,60,39)(3,95,49,40)(4,96,50,41)(5,85,51,42)(6,86,52,43)(7,87,53,44)(8,88,54,45)(9,89,55,46)(10,90,56,47)(11,91,57,48)(12,92,58,37)(13,65,82,29)(14,66,83,30)(15,67,84,31)(16,68,73,32)(17,69,74,33)(18,70,75,34)(19,71,76,35)(20,72,77,36)(21,61,78,25)(22,62,79,26)(23,63,80,27)(24,64,81,28), (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), (1,29,7,35)(2,28,8,34)(3,27,9,33)(4,26,10,32)(5,25,11,31)(6,36,12,30)(13,44,19,38)(14,43,20,37)(15,42,21,48)(16,41,22,47)(17,40,23,46)(18,39,24,45)(49,63,55,69)(50,62,56,68)(51,61,57,67)(52,72,58,66)(53,71,59,65)(54,70,60,64)(73,96,79,90)(74,95,80,89)(75,94,81,88)(76,93,82,87)(77,92,83,86)(78,91,84,85) );
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),(13,78,17,82,21,74),(14,79,18,83,22,75),(15,80,19,84,23,76),(16,81,20,73,24,77),(25,65,33,61,29,69),(26,66,34,62,30,70),(27,67,35,63,31,71),(28,68,36,64,32,72),(37,96,45,92,41,88),(38,85,46,93,42,89),(39,86,47,94,43,90),(40,87,48,95,44,91)], [(1,93,59,38),(2,94,60,39),(3,95,49,40),(4,96,50,41),(5,85,51,42),(6,86,52,43),(7,87,53,44),(8,88,54,45),(9,89,55,46),(10,90,56,47),(11,91,57,48),(12,92,58,37),(13,65,82,29),(14,66,83,30),(15,67,84,31),(16,68,73,32),(17,69,74,33),(18,70,75,34),(19,71,76,35),(20,72,77,36),(21,61,78,25),(22,62,79,26),(23,63,80,27),(24,64,81,28)], [(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)], [(1,29,7,35),(2,28,8,34),(3,27,9,33),(4,26,10,32),(5,25,11,31),(6,36,12,30),(13,44,19,38),(14,43,20,37),(15,42,21,48),(16,41,22,47),(17,40,23,46),(18,39,24,45),(49,63,55,69),(50,62,56,68),(51,61,57,67),(52,72,58,66),(53,71,59,65),(54,70,60,64),(73,96,79,90),(74,95,80,89),(75,94,81,88),(76,93,82,87),(77,92,83,86),(78,91,84,85)]])
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