direct product, metabelian, supersoluble, monomial
Aliases: Dic3×D12, C62.62C23, C12⋊1(C4×S3), C3⋊5(C4×D12), C32⋊5(C4×D4), (C3×D12)⋊7C4, C3⋊1(D4×Dic3), C4⋊1(S3×Dic3), C2.3(S3×D12), C6.16(S3×D4), D6⋊Dic3⋊3C2, (C3×Dic3)⋊8D4, (C4×Dic3)⋊5S3, D6⋊1(C2×Dic3), C12⋊4(C2×Dic3), C6.17(C2×D12), (C2×D12).10S3, (C6×D12).10C2, (C2×C12).136D6, (Dic3×C12)⋊10C2, C6.46(C4○D12), (C22×S3).38D6, C12⋊Dic3⋊18C2, C6.24(D4⋊2S3), C2.5(D12⋊5S3), (C6×C12).101C22, (C2×Dic3).113D6, C6.12(C22×Dic3), (C6×Dic3).111C22, (C2×C4).79S32, C6.91(S3×C2×C4), (S3×C6)⋊4(C2×C4), (C3×C12)⋊5(C2×C4), C22.35(C2×S32), (C2×S3×Dic3)⋊14C2, (C3×C6).49(C2×D4), C2.13(C2×S3×Dic3), (S3×C2×C6).21C22, (C3×C6).37(C4○D4), (C3×C6).56(C22×C4), (C2×C6).81(C22×S3), (C2×C3⋊Dic3).46C22, SmallGroup(288,540)
Series: Derived ►Chief ►Lower central ►Upper central
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, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3×C6, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C4×D4, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C62, C4×Dic3, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, S3×C2×C4, C2×D12, C22×Dic3, C6×D4, S3×Dic3, C3×D12, C6×Dic3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C4×D12, D4×Dic3, D6⋊Dic3, Dic3×C12, C12⋊Dic3, C2×S3×Dic3, C6×D12, Dic3×D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22×C4, C2×D4, C4○D4, C4×S3, D12, C2×Dic3, C22×S3, C4×D4, S32, S3×C2×C4, C2×D12, C4○D12, S3×D4, D4⋊2S3, C22×Dic3, S3×Dic3, C2×S32, C4×D12, D4×Dic3, D12⋊5S3, S3×D12, C2×S3×Dic3, Dic3×D12
►On 96 points
(1 33 9 29 5 25)(2 34 10 30 6 26)(3 35 11 31 7 27)(4 36 12 32 8 28)(13 89 21 85 17 93)(14 90 22 86 18 94)(15 91 23 87 19 95)(16 92 24 88 20 96)(37 60 41 52 45 56)(38 49 42 53 46 57)(39 50 43 54 47 58)(40 51 44 55 48 59)(61 84 65 76 69 80)(62 73 66 77 70 81)(63 74 67 78 71 82)(64 75 68 79 72 83)
(1 51 29 48)(2 52 30 37)(3 53 31 38)(4 54 32 39)(5 55 33 40)(6 56 34 41)(7 57 35 42)(8 58 36 43)(9 59 25 44)(10 60 26 45)(11 49 27 46)(12 50 28 47)(13 76 85 61)(14 77 86 62)(15 78 87 63)(16 79 88 64)(17 80 89 65)(18 81 90 66)(19 82 91 67)(20 83 92 68)(21 84 93 69)(22 73 94 70)(23 74 95 71)(24 75 96 72)
(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 84)(2 83)(3 82)(4 81)(5 80)(6 79)(7 78)(8 77)(9 76)(10 75)(11 74)(12 73)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 48)(22 47)(23 46)(24 45)(25 61)(26 72)(27 71)(28 70)(29 69)(30 68)(31 67)(32 66)(33 65)(34 64)(35 63)(36 62)(49 95)(50 94)(51 93)(52 92)(53 91)(54 90)(55 89)(56 88)(57 87)(58 86)(59 85)(60 96)
G:=sub<Sym(96)| (1,33,9,29,5,25)(2,34,10,30,6,26)(3,35,11,31,7,27)(4,36,12,32,8,28)(13,89,21,85,17,93)(14,90,22,86,18,94)(15,91,23,87,19,95)(16,92,24,88,20,96)(37,60,41,52,45,56)(38,49,42,53,46,57)(39,50,43,54,47,58)(40,51,44,55,48,59)(61,84,65,76,69,80)(62,73,66,77,70,81)(63,74,67,78,71,82)(64,75,68,79,72,83), (1,51,29,48)(2,52,30,37)(3,53,31,38)(4,54,32,39)(5,55,33,40)(6,56,34,41)(7,57,35,42)(8,58,36,43)(9,59,25,44)(10,60,26,45)(11,49,27,46)(12,50,28,47)(13,76,85,61)(14,77,86,62)(15,78,87,63)(16,79,88,64)(17,80,89,65)(18,81,90,66)(19,82,91,67)(20,83,92,68)(21,84,93,69)(22,73,94,70)(23,74,95,71)(24,75,96,72), (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), (1,84)(2,83)(3,82)(4,81)(5,80)(6,79)(7,78)(8,77)(9,76)(10,75)(11,74)(12,73)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,48)(22,47)(23,46)(24,45)(25,61)(26,72)(27,71)(28,70)(29,69)(30,68)(31,67)(32,66)(33,65)(34,64)(35,63)(36,62)(49,95)(50,94)(51,93)(52,92)(53,91)(54,90)(55,89)(56,88)(57,87)(58,86)(59,85)(60,96)>;
G:=Group( (1,33,9,29,5,25)(2,34,10,30,6,26)(3,35,11,31,7,27)(4,36,12,32,8,28)(13,89,21,85,17,93)(14,90,22,86,18,94)(15,91,23,87,19,95)(16,92,24,88,20,96)(37,60,41,52,45,56)(38,49,42,53,46,57)(39,50,43,54,47,58)(40,51,44,55,48,59)(61,84,65,76,69,80)(62,73,66,77,70,81)(63,74,67,78,71,82)(64,75,68,79,72,83), (1,51,29,48)(2,52,30,37)(3,53,31,38)(4,54,32,39)(5,55,33,40)(6,56,34,41)(7,57,35,42)(8,58,36,43)(9,59,25,44)(10,60,26,45)(11,49,27,46)(12,50,28,47)(13,76,85,61)(14,77,86,62)(15,78,87,63)(16,79,88,64)(17,80,89,65)(18,81,90,66)(19,82,91,67)(20,83,92,68)(21,84,93,69)(22,73,94,70)(23,74,95,71)(24,75,96,72), (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), (1,84)(2,83)(3,82)(4,81)(5,80)(6,79)(7,78)(8,77)(9,76)(10,75)(11,74)(12,73)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,48)(22,47)(23,46)(24,45)(25,61)(26,72)(27,71)(28,70)(29,69)(30,68)(31,67)(32,66)(33,65)(34,64)(35,63)(36,62)(49,95)(50,94)(51,93)(52,92)(53,91)(54,90)(55,89)(56,88)(57,87)(58,86)(59,85)(60,96) );
G=PermutationGroup([[(1,33,9,29,5,25),(2,34,10,30,6,26),(3,35,11,31,7,27),(4,36,12,32,8,28),(13,89,21,85,17,93),(14,90,22,86,18,94),(15,91,23,87,19,95),(16,92,24,88,20,96),(37,60,41,52,45,56),(38,49,42,53,46,57),(39,50,43,54,47,58),(40,51,44,55,48,59),(61,84,65,76,69,80),(62,73,66,77,70,81),(63,74,67,78,71,82),(64,75,68,79,72,83)], [(1,51,29,48),(2,52,30,37),(3,53,31,38),(4,54,32,39),(5,55,33,40),(6,56,34,41),(7,57,35,42),(8,58,36,43),(9,59,25,44),(10,60,26,45),(11,49,27,46),(12,50,28,47),(13,76,85,61),(14,77,86,62),(15,78,87,63),(16,79,88,64),(17,80,89,65),(18,81,90,66),(19,82,91,67),(20,83,92,68),(21,84,93,69),(22,73,94,70),(23,74,95,71),(24,75,96,72)], [(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)], [(1,84),(2,83),(3,82),(4,81),(5,80),(6,79),(7,78),(8,77),(9,76),(10,75),(11,74),(12,73),(13,44),(14,43),(15,42),(16,41),(17,40),(18,39),(19,38),(20,37),(21,48),(22,47),(23,46),(24,45),(25,61),(26,72),(27,71),(28,70),(29,69),(30,68),(31,67),(32,66),(33,65),(34,64),(35,63),(36,62),(49,95),(50,94),(51,93),(52,92),(53,91),(54,90),(55,89),(56,88),(57,87),(58,86),(59,85),(60,96)]])
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