metabelian, supersoluble, monomial
Aliases: Dic3⋊5Dic6, C62.7C23, C32⋊2(C4×Q8), C3⋊2(C4×Dic6), Dic32.8C2, C6.14(S3×Q8), (C3×Dic3)⋊5Q8, C32⋊2Q8⋊5C4, C6.1(C2×Dic6), C2.1(S3×Dic6), (C2×C12).185D6, Dic3⋊C4.9S3, (C4×Dic3).8S3, Dic3.3(C4×S3), C6.50(C4○D12), (Dic3×C12).2C2, (C2×Dic3).51D6, C3⋊1(Dic6⋊C4), C6.30(D4⋊2S3), Dic3⋊Dic3.4C2, (C6×C12).209C22, C2.2(D6.3D6), C6.Dic6.7C2, (C6×Dic3).29C22, C6.8(S3×C2×C4), C2.10(C4×S32), (C2×C4).38S32, (C3×C6).6(C2×Q8), C22.16(C2×S32), (C3×C6).8(C22×C4), (C3×C6).55(C4○D4), (C3×Dic3⋊C4).1C2, C3⋊Dic3.20(C2×C4), (C2×C6).26(C22×S3), (C2×C32⋊2Q8).6C2, (C3×Dic3).13(C2×C4), (C2×C3⋊Dic3).10C22, SmallGroup(288,485)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3⋊5Dic6
G = < a,b,c,d | a6=c12=1, b2=a3, d2=c6, bab-1=cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 458 in 153 conjugacy classes, 62 normal (44 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×11], C22, C6 [×6], C6 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C32, Dic3 [×6], Dic3 [×10], C12 [×10], C2×C6 [×2], C2×C6, C42 [×3], C4⋊C4 [×3], C2×Q8, C3×C6 [×3], Dic6 [×8], C2×Dic3 [×4], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×5], C4×Q8, C3×Dic3 [×6], C3×Dic3, C3⋊Dic3 [×2], C3⋊Dic3, C3×C12, C62, C4×Dic3, C4×Dic3 [×4], Dic3⋊C4, Dic3⋊C4 [×4], C4⋊Dic3, C4×C12, C3×C4⋊C4, C2×Dic6 [×2], C32⋊2Q8 [×4], C6×Dic3 [×4], C2×C3⋊Dic3 [×2], C6×C12, C4×Dic6, Dic6⋊C4, Dic32 [×2], Dic3⋊Dic3, Dic3×C12, C3×Dic3⋊C4, C6.Dic6, C2×C32⋊2Q8, Dic3⋊5Dic6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], Q8 [×2], C23, D6 [×6], C22×C4, C2×Q8, C4○D4, Dic6 [×2], C4×S3 [×4], C22×S3 [×2], C4×Q8, S32, C2×Dic6, S3×C2×C4 [×2], C4○D12, D4⋊2S3, S3×Q8, C2×S32, C4×Dic6, Dic6⋊C4, S3×Dic6, C4×S32, D6.3D6, Dic3⋊5Dic6
►On 96 points
(1 78 5 82 9 74)(2 75 10 83 6 79)(3 80 7 84 11 76)(4 77 12 73 8 81)(13 56 21 52 17 60)(14 49 18 53 22 57)(15 58 23 54 19 50)(16 51 20 55 24 59)(25 91 33 87 29 95)(26 96 30 88 34 92)(27 93 35 89 31 85)(28 86 32 90 36 94)(37 72 41 64 45 68)(38 69 46 65 42 61)(39 62 43 66 47 70)(40 71 48 67 44 63)
(1 60 82 21)(2 49 83 22)(3 50 84 23)(4 51 73 24)(5 52 74 13)(6 53 75 14)(7 54 76 15)(8 55 77 16)(9 56 78 17)(10 57 79 18)(11 58 80 19)(12 59 81 20)(25 66 87 39)(26 67 88 40)(27 68 89 41)(28 69 90 42)(29 70 91 43)(30 71 92 44)(31 72 93 45)(32 61 94 46)(33 62 95 47)(34 63 96 48)(35 64 85 37)(36 65 86 38)
(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 70 7 64)(2 69 8 63)(3 68 9 62)(4 67 10 61)(5 66 11 72)(6 65 12 71)(13 25 19 31)(14 36 20 30)(15 35 21 29)(16 34 22 28)(17 33 23 27)(18 32 24 26)(37 82 43 76)(38 81 44 75)(39 80 45 74)(40 79 46 73)(41 78 47 84)(42 77 48 83)(49 90 55 96)(50 89 56 95)(51 88 57 94)(52 87 58 93)(53 86 59 92)(54 85 60 91)
G:=sub<Sym(96)| (1,78,5,82,9,74)(2,75,10,83,6,79)(3,80,7,84,11,76)(4,77,12,73,8,81)(13,56,21,52,17,60)(14,49,18,53,22,57)(15,58,23,54,19,50)(16,51,20,55,24,59)(25,91,33,87,29,95)(26,96,30,88,34,92)(27,93,35,89,31,85)(28,86,32,90,36,94)(37,72,41,64,45,68)(38,69,46,65,42,61)(39,62,43,66,47,70)(40,71,48,67,44,63), (1,60,82,21)(2,49,83,22)(3,50,84,23)(4,51,73,24)(5,52,74,13)(6,53,75,14)(7,54,76,15)(8,55,77,16)(9,56,78,17)(10,57,79,18)(11,58,80,19)(12,59,81,20)(25,66,87,39)(26,67,88,40)(27,68,89,41)(28,69,90,42)(29,70,91,43)(30,71,92,44)(31,72,93,45)(32,61,94,46)(33,62,95,47)(34,63,96,48)(35,64,85,37)(36,65,86,38), (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), (1,70,7,64)(2,69,8,63)(3,68,9,62)(4,67,10,61)(5,66,11,72)(6,65,12,71)(13,25,19,31)(14,36,20,30)(15,35,21,29)(16,34,22,28)(17,33,23,27)(18,32,24,26)(37,82,43,76)(38,81,44,75)(39,80,45,74)(40,79,46,73)(41,78,47,84)(42,77,48,83)(49,90,55,96)(50,89,56,95)(51,88,57,94)(52,87,58,93)(53,86,59,92)(54,85,60,91)>;
G:=Group( (1,78,5,82,9,74)(2,75,10,83,6,79)(3,80,7,84,11,76)(4,77,12,73,8,81)(13,56,21,52,17,60)(14,49,18,53,22,57)(15,58,23,54,19,50)(16,51,20,55,24,59)(25,91,33,87,29,95)(26,96,30,88,34,92)(27,93,35,89,31,85)(28,86,32,90,36,94)(37,72,41,64,45,68)(38,69,46,65,42,61)(39,62,43,66,47,70)(40,71,48,67,44,63), (1,60,82,21)(2,49,83,22)(3,50,84,23)(4,51,73,24)(5,52,74,13)(6,53,75,14)(7,54,76,15)(8,55,77,16)(9,56,78,17)(10,57,79,18)(11,58,80,19)(12,59,81,20)(25,66,87,39)(26,67,88,40)(27,68,89,41)(28,69,90,42)(29,70,91,43)(30,71,92,44)(31,72,93,45)(32,61,94,46)(33,62,95,47)(34,63,96,48)(35,64,85,37)(36,65,86,38), (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), (1,70,7,64)(2,69,8,63)(3,68,9,62)(4,67,10,61)(5,66,11,72)(6,65,12,71)(13,25,19,31)(14,36,20,30)(15,35,21,29)(16,34,22,28)(17,33,23,27)(18,32,24,26)(37,82,43,76)(38,81,44,75)(39,80,45,74)(40,79,46,73)(41,78,47,84)(42,77,48,83)(49,90,55,96)(50,89,56,95)(51,88,57,94)(52,87,58,93)(53,86,59,92)(54,85,60,91) );
G=PermutationGroup([(1,78,5,82,9,74),(2,75,10,83,6,79),(3,80,7,84,11,76),(4,77,12,73,8,81),(13,56,21,52,17,60),(14,49,18,53,22,57),(15,58,23,54,19,50),(16,51,20,55,24,59),(25,91,33,87,29,95),(26,96,30,88,34,92),(27,93,35,89,31,85),(28,86,32,90,36,94),(37,72,41,64,45,68),(38,69,46,65,42,61),(39,62,43,66,47,70),(40,71,48,67,44,63)], [(1,60,82,21),(2,49,83,22),(3,50,84,23),(4,51,73,24),(5,52,74,13),(6,53,75,14),(7,54,76,15),(8,55,77,16),(9,56,78,17),(10,57,79,18),(11,58,80,19),(12,59,81,20),(25,66,87,39),(26,67,88,40),(27,68,89,41),(28,69,90,42),(29,70,91,43),(30,71,92,44),(31,72,93,45),(32,61,94,46),(33,62,95,47),(34,63,96,48),(35,64,85,37),(36,65,86,38)], [(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)], [(1,70,7,64),(2,69,8,63),(3,68,9,62),(4,67,10,61),(5,66,11,72),(6,65,12,71),(13,25,19,31),(14,36,20,30),(15,35,21,29),(16,34,22,28),(17,33,23,27),(18,32,24,26),(37,82,43,76),(38,81,44,75),(39,80,45,74),(40,79,46,73),(41,78,47,84),(42,77,48,83),(49,90,55,96),(50,89,56,95),(51,88,57,94),(52,87,58,93),(53,86,59,92),(54,85,60,91)])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | + | - | - | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | S3 | Q8 | D6 | D6 | C4○D4 | Dic6 | C4×S3 | C4○D12 | S32 | D4⋊2S3 | S3×Q8 | C2×S32 | S3×Dic6 | C4×S32 | D6.3D6 |
| kernel | Dic3⋊5Dic6 | Dic32 | Dic3⋊Dic3 | Dic3×C12 | C3×Dic3⋊C4 | C6.Dic6 | C2×C32⋊2Q8 | C32⋊2Q8 | C4×Dic3 | Dic3⋊C4 | C3×Dic3 | C2×Dic3 | C2×C12 | C3×C6 | Dic3 | Dic3 | C6 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 8 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of Dic3⋊5Dic6 ►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 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 8 | 11 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 3 | 5 | 0 | 0 | 0 | 0 |
| 11 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[8,0,0,0,0,0,11,5,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[3,11,0,0,0,0,5,10,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
Dic3⋊5Dic6 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes_5{\rm Dic}_6 % in TeX
G:=Group("Dic3:5Dic6"); // GroupNames label
G:=SmallGroup(288,485);
// by ID
G=gap.SmallGroup(288,485);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,590,100,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^12=1,b^2=a^3,d^2=c^6,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations