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