metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3:6SD16, C3:2(C4xSD16), D4.S3:2C4, D4.2(C4xS3), C6.32(C4xD4), C4:C4.130D6, Dic6:2(C2xC4), (C2xC8).198D6, C2.1(S3xSD16), (C8xDic3):18C2, D4:C4.9S3, (C2xD4).125D6, C6.20(C4oD8), C12.3(C22xC4), C12.Q8:1C2, (D4xDic3).2C2, C6.18(C2xSD16), C22.67(S3xD4), Dic6:C4:2C2, C2.1(D8:3S3), C2.Dic12:20C2, (C6xD4).19C22, C12.144(C4oD4), C4.45(D4:2S3), (C2xC12).198C23, (C2xC24).220C22, (C2xDic3).200D4, C4:Dic3.58C22, (C2xDic6).50C22, C2.16(Dic3:4D4), (C4xDic3).221C22, C4.3(S3xC2xC4), C3:C8:12(C2xC4), (C3xD4).2(C2xC4), (C2xC6).211(C2xD4), (C3xC4:C4).3C22, (C2xD4.S3).2C2, (C2xC3:C8).208C22, (C3xD4:C4).11C2, (C2xC4).305(C22xS3), SmallGroup(192,317)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3:6SD16
G = < a,b,c,d | a6=c8=d2=1, b2=a3, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c3 >
Subgroups: 312 in 122 conjugacy classes, 51 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2xC4, C2xC4, D4, D4, Q8, C23, Dic3, Dic3, C12, C12, C2xC6, C2xC6, C42, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, SD16, C22xC4, C2xD4, C2xQ8, C3:C8, C24, Dic6, Dic6, C2xDic3, C2xDic3, C2xC12, C2xC12, C3xD4, C3xD4, C22xC6, C4xC8, D4:C4, Q8:C4, C4.Q8, C4xD4, C4xQ8, C2xSD16, C2xC3:C8, C4xDic3, C4xDic3, Dic3:C4, C4:Dic3, D4.S3, C6.D4, C3xC4:C4, C2xC24, C2xDic6, C22xDic3, C6xD4, C4xSD16, C12.Q8, C8xDic3, C2.Dic12, C3xD4:C4, Dic6:C4, C2xD4.S3, D4xDic3, Dic3:6SD16
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, C23, D6, SD16, C22xC4, C2xD4, C4oD4, C4xS3, C22xS3, C4xD4, C2xSD16, C4oD8, S3xC2xC4, S3xD4, D4:2S3, C4xSD16, Dic3:4D4, D8:3S3, S3xSD16, Dic3:6SD16
►On 96 points
(1 88 51 16 41 78)(2 79 42 9 52 81)(3 82 53 10 43 80)(4 73 44 11 54 83)(5 84 55 12 45 74)(6 75 46 13 56 85)(7 86 49 14 47 76)(8 77 48 15 50 87)(17 89 35 59 32 71)(18 72 25 60 36 90)(19 91 37 61 26 65)(20 66 27 62 38 92)(21 93 39 63 28 67)(22 68 29 64 40 94)(23 95 33 57 30 69)(24 70 31 58 34 96)
(1 64 16 22)(2 57 9 23)(3 58 10 24)(4 59 11 17)(5 60 12 18)(6 61 13 19)(7 62 14 20)(8 63 15 21)(25 45 90 84)(26 46 91 85)(27 47 92 86)(28 48 93 87)(29 41 94 88)(30 42 95 81)(31 43 96 82)(32 44 89 83)(33 52 69 79)(34 53 70 80)(35 54 71 73)(36 55 72 74)(37 56 65 75)(38 49 66 76)(39 50 67 77)(40 51 68 78)
(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)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 23)(19 21)(20 24)(26 28)(27 31)(30 32)(33 35)(34 38)(37 39)(42 44)(43 47)(46 48)(49 53)(50 56)(52 54)(57 59)(58 62)(61 63)(65 67)(66 70)(69 71)(73 79)(75 77)(76 80)(81 83)(82 86)(85 87)(89 95)(91 93)(92 96)
G:=sub<Sym(96)| (1,88,51,16,41,78)(2,79,42,9,52,81)(3,82,53,10,43,80)(4,73,44,11,54,83)(5,84,55,12,45,74)(6,75,46,13,56,85)(7,86,49,14,47,76)(8,77,48,15,50,87)(17,89,35,59,32,71)(18,72,25,60,36,90)(19,91,37,61,26,65)(20,66,27,62,38,92)(21,93,39,63,28,67)(22,68,29,64,40,94)(23,95,33,57,30,69)(24,70,31,58,34,96), (1,64,16,22)(2,57,9,23)(3,58,10,24)(4,59,11,17)(5,60,12,18)(6,61,13,19)(7,62,14,20)(8,63,15,21)(25,45,90,84)(26,46,91,85)(27,47,92,86)(28,48,93,87)(29,41,94,88)(30,42,95,81)(31,43,96,82)(32,44,89,83)(33,52,69,79)(34,53,70,80)(35,54,71,73)(36,55,72,74)(37,56,65,75)(38,49,66,76)(39,50,67,77)(40,51,68,78), (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), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,23)(19,21)(20,24)(26,28)(27,31)(30,32)(33,35)(34,38)(37,39)(42,44)(43,47)(46,48)(49,53)(50,56)(52,54)(57,59)(58,62)(61,63)(65,67)(66,70)(69,71)(73,79)(75,77)(76,80)(81,83)(82,86)(85,87)(89,95)(91,93)(92,96)>;
G:=Group( (1,88,51,16,41,78)(2,79,42,9,52,81)(3,82,53,10,43,80)(4,73,44,11,54,83)(5,84,55,12,45,74)(6,75,46,13,56,85)(7,86,49,14,47,76)(8,77,48,15,50,87)(17,89,35,59,32,71)(18,72,25,60,36,90)(19,91,37,61,26,65)(20,66,27,62,38,92)(21,93,39,63,28,67)(22,68,29,64,40,94)(23,95,33,57,30,69)(24,70,31,58,34,96), (1,64,16,22)(2,57,9,23)(3,58,10,24)(4,59,11,17)(5,60,12,18)(6,61,13,19)(7,62,14,20)(8,63,15,21)(25,45,90,84)(26,46,91,85)(27,47,92,86)(28,48,93,87)(29,41,94,88)(30,42,95,81)(31,43,96,82)(32,44,89,83)(33,52,69,79)(34,53,70,80)(35,54,71,73)(36,55,72,74)(37,56,65,75)(38,49,66,76)(39,50,67,77)(40,51,68,78), (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), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,23)(19,21)(20,24)(26,28)(27,31)(30,32)(33,35)(34,38)(37,39)(42,44)(43,47)(46,48)(49,53)(50,56)(52,54)(57,59)(58,62)(61,63)(65,67)(66,70)(69,71)(73,79)(75,77)(76,80)(81,83)(82,86)(85,87)(89,95)(91,93)(92,96) );
G=PermutationGroup([[(1,88,51,16,41,78),(2,79,42,9,52,81),(3,82,53,10,43,80),(4,73,44,11,54,83),(5,84,55,12,45,74),(6,75,46,13,56,85),(7,86,49,14,47,76),(8,77,48,15,50,87),(17,89,35,59,32,71),(18,72,25,60,36,90),(19,91,37,61,26,65),(20,66,27,62,38,92),(21,93,39,63,28,67),(22,68,29,64,40,94),(23,95,33,57,30,69),(24,70,31,58,34,96)], [(1,64,16,22),(2,57,9,23),(3,58,10,24),(4,59,11,17),(5,60,12,18),(6,61,13,19),(7,62,14,20),(8,63,15,21),(25,45,90,84),(26,46,91,85),(27,47,92,86),(28,48,93,87),(29,41,94,88),(30,42,95,81),(31,43,96,82),(32,44,89,83),(33,52,69,79),(34,53,70,80),(35,54,71,73),(36,55,72,74),(37,56,65,75),(38,49,66,76),(39,50,67,77),(40,51,68,78)], [(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)], [(2,4),(3,7),(6,8),(9,11),(10,14),(13,15),(17,23),(19,21),(20,24),(26,28),(27,31),(30,32),(33,35),(34,38),(37,39),(42,44),(43,47),(46,48),(49,53),(50,56),(52,54),(57,59),(58,62),(61,63),(65,67),(66,70),(69,71),(73,79),(75,77),(76,80),(81,83),(82,86),(85,87),(89,95),(91,93),(92,96)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | D6 | SD16 | C4oD4 | C4xS3 | C4oD8 | D4:2S3 | S3xD4 | D8:3S3 | S3xSD16 |
| kernel | Dic3:6SD16 | C12.Q8 | C8xDic3 | C2.Dic12 | C3xD4:C4 | Dic6:C4 | C2xD4.S3 | D4xDic3 | D4.S3 | D4:C4 | C2xDic3 | C4:C4 | C2xC8 | C2xD4 | Dic3 | C12 | D4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of Dic3:6SD16 ►in GL4(F73) generated by
| 1 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 54 | 5 | 0 | 0 |
| 59 | 19 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 2 | 11 | 0 | 0 |
| 13 | 71 | 0 | 0 |
| 0 | 0 | 12 | 61 |
| 0 | 0 | 6 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 72 |
G:=sub<GL(4,GF(73))| [1,1,0,0,72,0,0,0,0,0,72,0,0,0,0,72],[54,59,0,0,5,19,0,0,0,0,27,0,0,0,0,27],[2,13,0,0,11,71,0,0,0,0,12,6,0,0,61,0],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,72] >;
Dic3:6SD16 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes_6{\rm SD}_{16} % in TeX
G:=Group("Dic3:6SD16"); // GroupNames label
G:=SmallGroup(192,317);
// by ID
G=gap.SmallGroup(192,317);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,135,268,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^8=d^2=1,b^2=a^3,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^3>;
// generators/relations