metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6:3Q16, C24.27D4, (C2xQ16):5S3, (C6xQ16):5C2, C24:1C4:24C2, (C2xC8).243D6, (C2xQ8).88D6, C6.30(C2xQ16), C2.19(S3xQ16), C6.81(C4oD8), C12.186(C2xD4), C8.28(C3:D4), C3:5(C8.18D4), D6:3Q8.8C2, Q8:2Dic3:35C2, (C2xC24).95C22, (C22xS3).60D4, C22.279(S3xD4), (C6xQ8).91C22, C12.107(C4oD4), C2.22(D6:3D4), C4.36(D4:2S3), C6.121(C4:D4), (C2xC12).462C23, (C2xDic3).117D4, C2.18(D24:C2), C4:Dic3.185C22, (S3xC2xC8).5C2, C4.85(C2xC3:D4), (C2xC6).373(C2xD4), (C2xC3:C8).278C22, (S3xC2xC4).241C22, (C2xC4).550(C22xS3), SmallGroup(192,747)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6:3Q16
G = < a,b,c,d | a6=b2=c8=1, d2=c4, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c-1 >
Subgroups: 296 in 114 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C8, C2xC4, C2xC4, Q8, C23, Dic3, C12, C12, D6, D6, C2xC6, C22:C4, C4:C4, C2xC8, C2xC8, Q16, C22xC4, C2xQ8, C3:C8, C24, C4xS3, C2xDic3, C2xDic3, C2xC12, C2xC12, C3xQ8, C22xS3, Q8:C4, C2.D8, C22:Q8, C22xC8, C2xQ16, S3xC8, C2xC3:C8, Dic3:C4, C4:Dic3, D6:C4, C2xC24, C3xQ16, S3xC2xC4, C6xQ8, C8.18D4, C24:1C4, Q8:2Dic3, S3xC2xC8, D6:3Q8, C6xQ16, D6:3Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2xD4, C4oD4, C3:D4, C22xS3, C4:D4, C2xQ16, C4oD8, S3xD4, D4:2S3, C2xC3:D4, C8.18D4, S3xQ16, D24:C2, D6:3D4, D6:3Q16
►On 96 points
Generators in S96
(1 31 79 45 34 95)(2 32 80 46 35 96)(3 25 73 47 36 89)(4 26 74 48 37 90)(5 27 75 41 38 91)(6 28 76 42 39 92)(7 29 77 43 40 93)(8 30 78 44 33 94)(9 61 84 70 24 49)(10 62 85 71 17 50)(11 63 86 72 18 51)(12 64 87 65 19 52)(13 57 88 66 20 53)(14 58 81 67 21 54)(15 59 82 68 22 55)(16 60 83 69 23 56)
(1 91)(2 92)(3 93)(4 94)(5 95)(6 96)(7 89)(8 90)(9 13)(10 14)(11 15)(12 16)(17 81)(18 82)(19 83)(20 84)(21 85)(22 86)(23 87)(24 88)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)(41 79)(42 80)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)(65 69)(66 70)(67 71)(68 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 88 5 84)(2 87 6 83)(3 86 7 82)(4 85 8 81)(9 34 13 38)(10 33 14 37)(11 40 15 36)(12 39 16 35)(17 78 21 74)(18 77 22 73)(19 76 23 80)(20 75 24 79)(25 72 29 68)(26 71 30 67)(27 70 31 66)(28 69 32 65)(41 49 45 53)(42 56 46 52)(43 55 47 51)(44 54 48 50)(57 91 61 95)(58 90 62 94)(59 89 63 93)(60 96 64 92)
G:=sub<Sym(96)| (1,31,79,45,34,95)(2,32,80,46,35,96)(3,25,73,47,36,89)(4,26,74,48,37,90)(5,27,75,41,38,91)(6,28,76,42,39,92)(7,29,77,43,40,93)(8,30,78,44,33,94)(9,61,84,70,24,49)(10,62,85,71,17,50)(11,63,86,72,18,51)(12,64,87,65,19,52)(13,57,88,66,20,53)(14,58,81,67,21,54)(15,59,82,68,22,55)(16,60,83,69,23,56), (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,89)(8,90)(9,13)(10,14)(11,15)(12,16)(17,81)(18,82)(19,83)(20,84)(21,85)(22,86)(23,87)(24,88)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(41,79)(42,80)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64)(65,69)(66,70)(67,71)(68,72), (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), (1,88,5,84)(2,87,6,83)(3,86,7,82)(4,85,8,81)(9,34,13,38)(10,33,14,37)(11,40,15,36)(12,39,16,35)(17,78,21,74)(18,77,22,73)(19,76,23,80)(20,75,24,79)(25,72,29,68)(26,71,30,67)(27,70,31,66)(28,69,32,65)(41,49,45,53)(42,56,46,52)(43,55,47,51)(44,54,48,50)(57,91,61,95)(58,90,62,94)(59,89,63,93)(60,96,64,92)>;
G:=Group( (1,31,79,45,34,95)(2,32,80,46,35,96)(3,25,73,47,36,89)(4,26,74,48,37,90)(5,27,75,41,38,91)(6,28,76,42,39,92)(7,29,77,43,40,93)(8,30,78,44,33,94)(9,61,84,70,24,49)(10,62,85,71,17,50)(11,63,86,72,18,51)(12,64,87,65,19,52)(13,57,88,66,20,53)(14,58,81,67,21,54)(15,59,82,68,22,55)(16,60,83,69,23,56), (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,89)(8,90)(9,13)(10,14)(11,15)(12,16)(17,81)(18,82)(19,83)(20,84)(21,85)(22,86)(23,87)(24,88)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(41,79)(42,80)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64)(65,69)(66,70)(67,71)(68,72), (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), (1,88,5,84)(2,87,6,83)(3,86,7,82)(4,85,8,81)(9,34,13,38)(10,33,14,37)(11,40,15,36)(12,39,16,35)(17,78,21,74)(18,77,22,73)(19,76,23,80)(20,75,24,79)(25,72,29,68)(26,71,30,67)(27,70,31,66)(28,69,32,65)(41,49,45,53)(42,56,46,52)(43,55,47,51)(44,54,48,50)(57,91,61,95)(58,90,62,94)(59,89,63,93)(60,96,64,92) );
G=PermutationGroup([[(1,31,79,45,34,95),(2,32,80,46,35,96),(3,25,73,47,36,89),(4,26,74,48,37,90),(5,27,75,41,38,91),(6,28,76,42,39,92),(7,29,77,43,40,93),(8,30,78,44,33,94),(9,61,84,70,24,49),(10,62,85,71,17,50),(11,63,86,72,18,51),(12,64,87,65,19,52),(13,57,88,66,20,53),(14,58,81,67,21,54),(15,59,82,68,22,55),(16,60,83,69,23,56)], [(1,91),(2,92),(3,93),(4,94),(5,95),(6,96),(7,89),(8,90),(9,13),(10,14),(11,15),(12,16),(17,81),(18,82),(19,83),(20,84),(21,85),(22,86),(23,87),(24,88),(25,40),(26,33),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39),(41,79),(42,80),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63),(56,64),(65,69),(66,70),(67,71),(68,72)], [(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)], [(1,88,5,84),(2,87,6,83),(3,86,7,82),(4,85,8,81),(9,34,13,38),(10,33,14,37),(11,40,15,36),(12,39,16,35),(17,78,21,74),(18,77,22,73),(19,76,23,80),(20,75,24,79),(25,72,29,68),(26,71,30,67),(27,70,31,66),(28,69,32,65),(41,49,45,53),(42,56,46,52),(43,55,47,51),(44,54,48,50),(57,91,61,95),(58,90,62,94),(59,89,63,93),(60,96,64,92)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 6 | 6 | 8 | 8 | 24 | 24 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | C4oD4 | Q16 | C3:D4 | C4oD8 | D4:2S3 | S3xD4 | S3xQ16 | D24:C2 |
| kernel | D6:3Q16 | C24:1C4 | Q8:2Dic3 | S3xC2xC8 | D6:3Q8 | C6xQ16 | C2xQ16 | C24 | C2xDic3 | C22xS3 | C2xC8 | C2xQ8 | C12 | D6 | C8 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of D6:3Q16 ►in GL6(F73)
| 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 29 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 0 | 0 | 0 |
| 0 | 0 | 28 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 63 | 0 |
| 0 | 0 | 0 | 0 | 41 | 51 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 67 | 0 | 0 |
| 0 | 0 | 45 | 59 | 0 | 0 |
| 0 | 0 | 0 | 0 | 57 | 67 |
| 0 | 0 | 0 | 0 | 55 | 16 |
G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,72,1,0,0,0,0,0,0,1,29,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,22,28,0,0,0,0,0,10,0,0,0,0,0,0,63,41,0,0,0,0,0,51],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,14,45,0,0,0,0,67,59,0,0,0,0,0,0,57,55,0,0,0,0,67,16] >;
D6:3Q16 in GAP, Magma, Sage, TeX
D_6\rtimes_3Q_{16} % in TeX
G:=Group("D6:3Q16"); // GroupNames label
G:=SmallGroup(192,747);
// by ID
G=gap.SmallGroup(192,747);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,184,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^8=1,d^2=c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations