metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D16:3S3, D6.1D8, D8.1D6, C16.8D6, Dic24:4C2, C48.6C22, C24.15C23, Dic3.12D8, Dic12.2C22, C3:C8.12D4, C4.3(S3xD4), (S3xC16):2C2, (C3xD16):3C2, C3:2(C4oD16), D8.S3:2C2, C6.34(C2xD8), C12.9(C2xD4), C2.18(S3xD8), D8:3S3:4C2, (C4xS3).19D4, C3:C16.5C22, C8.21(C22xS3), (C3xD8).1C22, (S3xC8).10C22, SmallGroup(192,471)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D16:3S3
G = < a,b,c,d | a16=b2=c3=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a8b, dcd=c-1 >
Subgroups: 284 in 84 conjugacy classes, 31 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2xC4, D4, Q8, Dic3, Dic3, C12, D6, C2xC6, C16, C16, C2xC8, D8, SD16, Q16, C4oD4, C3:C8, C24, Dic6, C4xS3, C2xDic3, C3:D4, C3xD4, C2xC16, D16, SD32, Q32, C4oD8, C3:C16, C48, S3xC8, Dic12, D4.S3, C3xD8, D4:2S3, C4oD16, S3xC16, Dic24, D8.S3, C3xD16, D8:3S3, D16:3S3
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2xD4, C22xS3, C2xD8, S3xD4, C4oD16, S3xD8, D16:3S3
►On 96 points
Generators in S96
(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 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 19)(20 32)(21 31)(22 30)(23 29)(24 28)(25 27)(33 35)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)(49 51)(52 64)(53 63)(54 62)(55 61)(56 60)(57 59)(65 73)(66 72)(67 71)(68 70)(74 80)(75 79)(76 78)(81 85)(82 84)(86 96)(87 95)(88 94)(89 93)(90 92)
(1 18 58)(2 19 59)(3 20 60)(4 21 61)(5 22 62)(6 23 63)(7 24 64)(8 25 49)(9 26 50)(10 27 51)(11 28 52)(12 29 53)(13 30 54)(14 31 55)(15 32 56)(16 17 57)(33 82 76)(34 83 77)(35 84 78)(36 85 79)(37 86 80)(38 87 65)(39 88 66)(40 89 67)(41 90 68)(42 91 69)(43 92 70)(44 93 71)(45 94 72)(46 95 73)(47 96 74)(48 81 75)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 94)(18 95)(19 96)(20 81)(21 82)(22 83)(23 84)(24 85)(25 86)(26 87)(27 88)(28 89)(29 90)(30 91)(31 92)(32 93)(33 61)(34 62)(35 63)(36 64)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)
G:=sub<Sym(96)| (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,19)(20,32)(21,31)(22,30)(23,29)(24,28)(25,27)(33,35)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(49,51)(52,64)(53,63)(54,62)(55,61)(56,60)(57,59)(65,73)(66,72)(67,71)(68,70)(74,80)(75,79)(76,78)(81,85)(82,84)(86,96)(87,95)(88,94)(89,93)(90,92), (1,18,58)(2,19,59)(3,20,60)(4,21,61)(5,22,62)(6,23,63)(7,24,64)(8,25,49)(9,26,50)(10,27,51)(11,28,52)(12,29,53)(13,30,54)(14,31,55)(15,32,56)(16,17,57)(33,82,76)(34,83,77)(35,84,78)(36,85,79)(37,86,80)(38,87,65)(39,88,66)(40,89,67)(41,90,68)(42,91,69)(43,92,70)(44,93,71)(45,94,72)(46,95,73)(47,96,74)(48,81,75), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,94)(18,95)(19,96)(20,81)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,91)(31,92)(32,93)(33,61)(34,62)(35,63)(36,64)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60)>;
G:=Group( (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,19)(20,32)(21,31)(22,30)(23,29)(24,28)(25,27)(33,35)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(49,51)(52,64)(53,63)(54,62)(55,61)(56,60)(57,59)(65,73)(66,72)(67,71)(68,70)(74,80)(75,79)(76,78)(81,85)(82,84)(86,96)(87,95)(88,94)(89,93)(90,92), (1,18,58)(2,19,59)(3,20,60)(4,21,61)(5,22,62)(6,23,63)(7,24,64)(8,25,49)(9,26,50)(10,27,51)(11,28,52)(12,29,53)(13,30,54)(14,31,55)(15,32,56)(16,17,57)(33,82,76)(34,83,77)(35,84,78)(36,85,79)(37,86,80)(38,87,65)(39,88,66)(40,89,67)(41,90,68)(42,91,69)(43,92,70)(44,93,71)(45,94,72)(46,95,73)(47,96,74)(48,81,75), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,94)(18,95)(19,96)(20,81)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,91)(31,92)(32,93)(33,61)(34,62)(35,63)(36,64)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60) );
G=PermutationGroup([[(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)], [(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(17,19),(20,32),(21,31),(22,30),(23,29),(24,28),(25,27),(33,35),(36,48),(37,47),(38,46),(39,45),(40,44),(41,43),(49,51),(52,64),(53,63),(54,62),(55,61),(56,60),(57,59),(65,73),(66,72),(67,71),(68,70),(74,80),(75,79),(76,78),(81,85),(82,84),(86,96),(87,95),(88,94),(89,93),(90,92)], [(1,18,58),(2,19,59),(3,20,60),(4,21,61),(5,22,62),(6,23,63),(7,24,64),(8,25,49),(9,26,50),(10,27,51),(11,28,52),(12,29,53),(13,30,54),(14,31,55),(15,32,56),(16,17,57),(33,82,76),(34,83,77),(35,84,78),(36,85,79),(37,86,80),(38,87,65),(39,88,66),(40,89,67),(41,90,68),(42,91,69),(43,92,70),(44,93,71),(45,94,72),(46,95,73),(47,96,74),(48,81,75)], [(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,79),(8,80),(9,65),(10,66),(11,67),(12,68),(13,69),(14,70),(15,71),(16,72),(17,94),(18,95),(19,96),(20,81),(21,82),(22,83),(23,84),(24,85),(25,86),(26,87),(27,88),(28,89),(29,90),(30,91),(31,92),(32,93),(33,61),(34,62),(35,63),(36,64),(37,49),(38,50),(39,51),(40,52),(41,53),(42,54),(43,55),(44,56),(45,57),(46,58),(47,59),(48,60)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12 | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 24A | 24B | 48A | 48B | 48C | 48D |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 24 | 24 | 48 | 48 | 48 | 48 |
| size | 1 | 1 | 6 | 8 | 8 | 2 | 2 | 3 | 3 | 24 | 24 | 2 | 16 | 16 | 2 | 2 | 6 | 6 | 4 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D8 | D8 | C4oD16 | S3xD4 | S3xD8 | D16:3S3 |
| kernel | D16:3S3 | S3xC16 | Dic24 | D8.S3 | C3xD16 | D8:3S3 | D16 | C3:C8 | C4xS3 | C16 | D8 | Dic3 | D6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 8 | 1 | 2 | 4 |
Matrix representation of D16:3S3 ►in GL4(F97) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 24 | 91 |
| 0 | 0 | 66 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 64 | 96 |
| 96 | 96 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 44 | 20 | 0 | 0 |
| 73 | 53 | 0 | 0 |
| 0 | 0 | 22 | 66 |
| 0 | 0 | 50 | 75 |
G:=sub<GL(4,GF(97))| [1,0,0,0,0,1,0,0,0,0,24,66,0,0,91,28],[1,0,0,0,0,1,0,0,0,0,1,64,0,0,0,96],[96,1,0,0,96,0,0,0,0,0,1,0,0,0,0,1],[44,73,0,0,20,53,0,0,0,0,22,50,0,0,66,75] >;
D16:3S3 in GAP, Magma, Sage, TeX
D_{16}\rtimes_3S_3 % in TeX
G:=Group("D16:3S3"); // GroupNames label
G:=SmallGroup(192,471);
// by ID
G=gap.SmallGroup(192,471);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,758,135,346,185,192,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^16=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^8*b,d*c*d=c^-1>;
// generators/relations