direct product, metabelian, supersoluble, monomial
Aliases: Dic3×Dic6, C62.12C23, C32⋊4(C4×Q8), C3⋊5(C4×Dic6), C3⋊1(Q8×Dic3), Dic32.4C2, C6.16(S3×Q8), C12.33(C4×S3), (C3×Dic3)⋊6Q8, (C3×Dic6)⋊7C4, C4.5(S3×Dic3), C2.2(S3×Dic6), C6.4(C2×Dic6), C6.3(C4○D12), (C2×C12).126D6, (C4×Dic3).1S3, (C6×Dic6).9C2, (C6×C12).86C22, C6.3(Q8⋊3S3), (Dic3×C12).5C2, (C2×Dic3).54D6, (C2×Dic6).10S3, C12.26(C2×Dic3), C6.7(C22×Dic3), Dic3⋊Dic3.8C2, Dic3.3(C2×Dic3), C2.3(D6.6D6), C12⋊Dic3.15C2, (C6×Dic3).105C22, (C2×C4).70S32, C6.87(S3×C2×C4), C2.9(C2×S3×Dic3), C22.19(C2×S32), (C3×C6).10(C2×Q8), (C3×C12).61(C2×C4), (C3×C6).3(C4○D4), (C2×C6).31(C22×S3), (C3×C6).47(C22×C4), (C3×Dic3).8(C2×C4), (C2×C3⋊Dic3).14C22, SmallGroup(288,490)
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, bd=db, dcd-1=c-1 >
Subgroups: 426 in 153 conjugacy classes, 72 normal (34 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, Q8, C32, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C2×Q8, C3×C6, Dic6, C2×Dic3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C4×Q8, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C62, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×Dic6, C6×Q8, C3×Dic6, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C6×C12, C4×Dic6, Q8×Dic3, Dic32, Dic3⋊Dic3, Dic3×C12, C12⋊Dic3, C6×Dic6, Dic3×Dic6
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, Dic3, D6, C22×C4, C2×Q8, C4○D4, Dic6, C4×S3, C2×Dic3, C22×S3, C4×Q8, S32, C2×Dic6, S3×C2×C4, C4○D12, S3×Q8, Q8⋊3S3, C22×Dic3, S3×Dic3, C2×S32, C4×Dic6, Q8×Dic3, S3×Dic6, D6.6D6, C2×S3×Dic3, Dic3×Dic6
►On 96 points
(1 27 5 31 9 35)(2 28 6 32 10 36)(3 29 7 33 11 25)(4 30 8 34 12 26)(13 85 21 93 17 89)(14 86 22 94 18 90)(15 87 23 95 19 91)(16 88 24 96 20 92)(37 59 41 51 45 55)(38 60 42 52 46 56)(39 49 43 53 47 57)(40 50 44 54 48 58)(61 80 69 76 65 84)(62 81 70 77 66 73)(63 82 71 78 67 74)(64 83 72 79 68 75)
(1 72 31 75)(2 61 32 76)(3 62 33 77)(4 63 34 78)(5 64 35 79)(6 65 36 80)(7 66 25 81)(8 67 26 82)(9 68 27 83)(10 69 28 84)(11 70 29 73)(12 71 30 74)(13 50 93 48)(14 51 94 37)(15 52 95 38)(16 53 96 39)(17 54 85 40)(18 55 86 41)(19 56 87 42)(20 57 88 43)(21 58 89 44)(22 59 90 45)(23 60 91 46)(24 49 92 47)
(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 19 7 13)(2 18 8 24)(3 17 9 23)(4 16 10 22)(5 15 11 21)(6 14 12 20)(25 93 31 87)(26 92 32 86)(27 91 33 85)(28 90 34 96)(29 89 35 95)(30 88 36 94)(37 74 43 80)(38 73 44 79)(39 84 45 78)(40 83 46 77)(41 82 47 76)(42 81 48 75)(49 61 55 67)(50 72 56 66)(51 71 57 65)(52 70 58 64)(53 69 59 63)(54 68 60 62)
G:=sub<Sym(96)| (1,27,5,31,9,35)(2,28,6,32,10,36)(3,29,7,33,11,25)(4,30,8,34,12,26)(13,85,21,93,17,89)(14,86,22,94,18,90)(15,87,23,95,19,91)(16,88,24,96,20,92)(37,59,41,51,45,55)(38,60,42,52,46,56)(39,49,43,53,47,57)(40,50,44,54,48,58)(61,80,69,76,65,84)(62,81,70,77,66,73)(63,82,71,78,67,74)(64,83,72,79,68,75), (1,72,31,75)(2,61,32,76)(3,62,33,77)(4,63,34,78)(5,64,35,79)(6,65,36,80)(7,66,25,81)(8,67,26,82)(9,68,27,83)(10,69,28,84)(11,70,29,73)(12,71,30,74)(13,50,93,48)(14,51,94,37)(15,52,95,38)(16,53,96,39)(17,54,85,40)(18,55,86,41)(19,56,87,42)(20,57,88,43)(21,58,89,44)(22,59,90,45)(23,60,91,46)(24,49,92,47), (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), (1,19,7,13)(2,18,8,24)(3,17,9,23)(4,16,10,22)(5,15,11,21)(6,14,12,20)(25,93,31,87)(26,92,32,86)(27,91,33,85)(28,90,34,96)(29,89,35,95)(30,88,36,94)(37,74,43,80)(38,73,44,79)(39,84,45,78)(40,83,46,77)(41,82,47,76)(42,81,48,75)(49,61,55,67)(50,72,56,66)(51,71,57,65)(52,70,58,64)(53,69,59,63)(54,68,60,62)>;
G:=Group( (1,27,5,31,9,35)(2,28,6,32,10,36)(3,29,7,33,11,25)(4,30,8,34,12,26)(13,85,21,93,17,89)(14,86,22,94,18,90)(15,87,23,95,19,91)(16,88,24,96,20,92)(37,59,41,51,45,55)(38,60,42,52,46,56)(39,49,43,53,47,57)(40,50,44,54,48,58)(61,80,69,76,65,84)(62,81,70,77,66,73)(63,82,71,78,67,74)(64,83,72,79,68,75), (1,72,31,75)(2,61,32,76)(3,62,33,77)(4,63,34,78)(5,64,35,79)(6,65,36,80)(7,66,25,81)(8,67,26,82)(9,68,27,83)(10,69,28,84)(11,70,29,73)(12,71,30,74)(13,50,93,48)(14,51,94,37)(15,52,95,38)(16,53,96,39)(17,54,85,40)(18,55,86,41)(19,56,87,42)(20,57,88,43)(21,58,89,44)(22,59,90,45)(23,60,91,46)(24,49,92,47), (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), (1,19,7,13)(2,18,8,24)(3,17,9,23)(4,16,10,22)(5,15,11,21)(6,14,12,20)(25,93,31,87)(26,92,32,86)(27,91,33,85)(28,90,34,96)(29,89,35,95)(30,88,36,94)(37,74,43,80)(38,73,44,79)(39,84,45,78)(40,83,46,77)(41,82,47,76)(42,81,48,75)(49,61,55,67)(50,72,56,66)(51,71,57,65)(52,70,58,64)(53,69,59,63)(54,68,60,62) );
G=PermutationGroup([[(1,27,5,31,9,35),(2,28,6,32,10,36),(3,29,7,33,11,25),(4,30,8,34,12,26),(13,85,21,93,17,89),(14,86,22,94,18,90),(15,87,23,95,19,91),(16,88,24,96,20,92),(37,59,41,51,45,55),(38,60,42,52,46,56),(39,49,43,53,47,57),(40,50,44,54,48,58),(61,80,69,76,65,84),(62,81,70,77,66,73),(63,82,71,78,67,74),(64,83,72,79,68,75)], [(1,72,31,75),(2,61,32,76),(3,62,33,77),(4,63,34,78),(5,64,35,79),(6,65,36,80),(7,66,25,81),(8,67,26,82),(9,68,27,83),(10,69,28,84),(11,70,29,73),(12,71,30,74),(13,50,93,48),(14,51,94,37),(15,52,95,38),(16,53,96,39),(17,54,85,40),(18,55,86,41),(19,56,87,42),(20,57,88,43),(21,58,89,44),(22,59,90,45),(23,60,91,46),(24,49,92,47)], [(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)], [(1,19,7,13),(2,18,8,24),(3,17,9,23),(4,16,10,22),(5,15,11,21),(6,14,12,20),(25,93,31,87),(26,92,32,86),(27,91,33,85),(28,90,34,96),(29,89,35,95),(30,88,36,94),(37,74,43,80),(38,73,44,79),(39,84,45,78),(40,83,46,77),(41,82,47,76),(42,81,48,75),(49,61,55,67),(50,72,56,66),(51,71,57,65),(52,70,58,64),(53,69,59,63),(54,68,60,62)]])
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