direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3×D8, C3⋊5(C4×D8), C24⋊6(C2×C4), (C3×D8)⋊2C4, C2.5(S3×D8), (C6×D8).4C2, (C2×D8).7S3, C8⋊4(C2×Dic3), C6.93(C4×D4), C6.42(C2×D8), (C8×Dic3)⋊4C2, (D4×Dic3)⋊3C2, D4⋊1(C2×Dic3), C24⋊1C4⋊22C2, (C2×C8).235D6, (C2×D4).139D6, C6.31(C4○D8), C2.10(D4×Dic3), C12.89(C4○D4), D4⋊Dic3⋊24C2, C2.5(D8⋊3S3), (C2×C24).87C22, C12.71(C22×C4), (C6×D4).76C22, C22.114(S3×D4), C4.26(D4⋊2S3), C4.1(C22×Dic3), (C2×C12).426C23, (C2×Dic3).212D4, C4⋊Dic3.161C22, (C4×Dic3).237C22, (C3×D4)⋊6(C2×C4), (C2×C6).339(C2×D4), (C2×C3⋊C8).268C22, (C2×C4).516(C22×S3), SmallGroup(192,708)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3×D8
G = < a,b,c,d | a6=c8=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 360 in 134 conjugacy classes, 59 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, C23, Dic3, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C3⋊C8, C24, C2×Dic3, C2×Dic3, C2×C12, C3×D4, C3×D4, C22×C6, C4×C8, D4⋊C4, C2.D8, C4×D4, C2×D8, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, C6.D4, C2×C24, C3×D8, C22×Dic3, C6×D4, C4×D8, C8×Dic3, C24⋊1C4, D4⋊Dic3, D4×Dic3, C6×D8, Dic3×D8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, D8, C22×C4, C2×D4, C4○D4, C2×Dic3, C22×S3, C4×D4, C2×D8, C4○D8, S3×D4, D4⋊2S3, C22×Dic3, C4×D8, S3×D8, D8⋊3S3, D4×Dic3, Dic3×D8
►On 96 points
(1 26 75 19 86 66)(2 27 76 20 87 67)(3 28 77 21 88 68)(4 29 78 22 81 69)(5 30 79 23 82 70)(6 31 80 24 83 71)(7 32 73 17 84 72)(8 25 74 18 85 65)(9 41 57 53 94 38)(10 42 58 54 95 39)(11 43 59 55 96 40)(12 44 60 56 89 33)(13 45 61 49 90 34)(14 46 62 50 91 35)(15 47 63 51 92 36)(16 48 64 52 93 37)
(1 57 19 38)(2 58 20 39)(3 59 21 40)(4 60 22 33)(5 61 23 34)(6 62 24 35)(7 63 17 36)(8 64 18 37)(9 66 53 75)(10 67 54 76)(11 68 55 77)(12 69 56 78)(13 70 49 79)(14 71 50 80)(15 72 51 73)(16 65 52 74)(25 48 85 93)(26 41 86 94)(27 42 87 95)(28 43 88 96)(29 44 81 89)(30 45 82 90)(31 46 83 91)(32 47 84 92)
(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 18)(2 17)(3 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 52)(10 51)(11 50)(12 49)(13 56)(14 55)(15 54)(16 53)(25 86)(26 85)(27 84)(28 83)(29 82)(30 81)(31 88)(32 87)(33 61)(34 60)(35 59)(36 58)(37 57)(38 64)(39 63)(40 62)(41 93)(42 92)(43 91)(44 90)(45 89)(46 96)(47 95)(48 94)(65 75)(66 74)(67 73)(68 80)(69 79)(70 78)(71 77)(72 76)
G:=sub<Sym(96)| (1,26,75,19,86,66)(2,27,76,20,87,67)(3,28,77,21,88,68)(4,29,78,22,81,69)(5,30,79,23,82,70)(6,31,80,24,83,71)(7,32,73,17,84,72)(8,25,74,18,85,65)(9,41,57,53,94,38)(10,42,58,54,95,39)(11,43,59,55,96,40)(12,44,60,56,89,33)(13,45,61,49,90,34)(14,46,62,50,91,35)(15,47,63,51,92,36)(16,48,64,52,93,37), (1,57,19,38)(2,58,20,39)(3,59,21,40)(4,60,22,33)(5,61,23,34)(6,62,24,35)(7,63,17,36)(8,64,18,37)(9,66,53,75)(10,67,54,76)(11,68,55,77)(12,69,56,78)(13,70,49,79)(14,71,50,80)(15,72,51,73)(16,65,52,74)(25,48,85,93)(26,41,86,94)(27,42,87,95)(28,43,88,96)(29,44,81,89)(30,45,82,90)(31,46,83,91)(32,47,84,92), (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), (1,18)(2,17)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,52)(10,51)(11,50)(12,49)(13,56)(14,55)(15,54)(16,53)(25,86)(26,85)(27,84)(28,83)(29,82)(30,81)(31,88)(32,87)(33,61)(34,60)(35,59)(36,58)(37,57)(38,64)(39,63)(40,62)(41,93)(42,92)(43,91)(44,90)(45,89)(46,96)(47,95)(48,94)(65,75)(66,74)(67,73)(68,80)(69,79)(70,78)(71,77)(72,76)>;
G:=Group( (1,26,75,19,86,66)(2,27,76,20,87,67)(3,28,77,21,88,68)(4,29,78,22,81,69)(5,30,79,23,82,70)(6,31,80,24,83,71)(7,32,73,17,84,72)(8,25,74,18,85,65)(9,41,57,53,94,38)(10,42,58,54,95,39)(11,43,59,55,96,40)(12,44,60,56,89,33)(13,45,61,49,90,34)(14,46,62,50,91,35)(15,47,63,51,92,36)(16,48,64,52,93,37), (1,57,19,38)(2,58,20,39)(3,59,21,40)(4,60,22,33)(5,61,23,34)(6,62,24,35)(7,63,17,36)(8,64,18,37)(9,66,53,75)(10,67,54,76)(11,68,55,77)(12,69,56,78)(13,70,49,79)(14,71,50,80)(15,72,51,73)(16,65,52,74)(25,48,85,93)(26,41,86,94)(27,42,87,95)(28,43,88,96)(29,44,81,89)(30,45,82,90)(31,46,83,91)(32,47,84,92), (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), (1,18)(2,17)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,52)(10,51)(11,50)(12,49)(13,56)(14,55)(15,54)(16,53)(25,86)(26,85)(27,84)(28,83)(29,82)(30,81)(31,88)(32,87)(33,61)(34,60)(35,59)(36,58)(37,57)(38,64)(39,63)(40,62)(41,93)(42,92)(43,91)(44,90)(45,89)(46,96)(47,95)(48,94)(65,75)(66,74)(67,73)(68,80)(69,79)(70,78)(71,77)(72,76) );
G=PermutationGroup([[(1,26,75,19,86,66),(2,27,76,20,87,67),(3,28,77,21,88,68),(4,29,78,22,81,69),(5,30,79,23,82,70),(6,31,80,24,83,71),(7,32,73,17,84,72),(8,25,74,18,85,65),(9,41,57,53,94,38),(10,42,58,54,95,39),(11,43,59,55,96,40),(12,44,60,56,89,33),(13,45,61,49,90,34),(14,46,62,50,91,35),(15,47,63,51,92,36),(16,48,64,52,93,37)], [(1,57,19,38),(2,58,20,39),(3,59,21,40),(4,60,22,33),(5,61,23,34),(6,62,24,35),(7,63,17,36),(8,64,18,37),(9,66,53,75),(10,67,54,76),(11,68,55,77),(12,69,56,78),(13,70,49,79),(14,71,50,80),(15,72,51,73),(16,65,52,74),(25,48,85,93),(26,41,86,94),(27,42,87,95),(28,43,88,96),(29,44,81,89),(30,45,82,90),(31,46,83,91),(32,47,84,92)], [(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)], [(1,18),(2,17),(3,24),(4,23),(5,22),(6,21),(7,20),(8,19),(9,52),(10,51),(11,50),(12,49),(13,56),(14,55),(15,54),(16,53),(25,86),(26,85),(27,84),(28,83),(29,82),(30,81),(31,88),(32,87),(33,61),(34,60),(35,59),(36,58),(37,57),(38,64),(39,63),(40,62),(41,93),(42,92),(43,91),(44,90),(45,89),(46,96),(47,95),(48,94),(65,75),(66,74),(67,73),(68,80),(69,79),(70,78),(71,77),(72,76)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 4 | 4 | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | Dic3 | D6 | D8 | C4○D4 | C4○D8 | D4⋊2S3 | S3×D4 | S3×D8 | D8⋊3S3 |
| kernel | Dic3×D8 | C8×Dic3 | C24⋊1C4 | D4⋊Dic3 | D4×Dic3 | C6×D8 | C3×D8 | C2×D8 | C2×Dic3 | C2×C8 | D8 | C2×D4 | Dic3 | C12 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 8 | 1 | 2 | 1 | 4 | 2 | 4 | 2 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of Dic3×D8 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 72 |
| 0 | 0 | 1 | 0 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 53 | 18 |
| 0 | 0 | 71 | 20 |
| 57 | 16 | 0 | 0 |
| 57 | 57 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 16 | 57 | 0 | 0 |
| 57 | 57 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,1,0,0,72,0],[46,0,0,0,0,46,0,0,0,0,53,71,0,0,18,20],[57,57,0,0,16,57,0,0,0,0,72,0,0,0,0,72],[16,57,0,0,57,57,0,0,0,0,72,0,0,0,0,72] >;
Dic3×D8 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times D_8 % in TeX
G:=Group("Dic3xD8"); // GroupNames label
G:=SmallGroup(192,708);
// by ID
G=gap.SmallGroup(192,708);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,219,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^8=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