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## G = D16.S3order 192 = 26·3

### The non-split extension by D16 of S3 acting via S3/C3=C2

Aliases: D16.S3, C32SD64, C6.9D16, C16.5D6, C12.6D8, C24.10D4, Dic243C2, C48.3C22, C3⋊C322C2, C4.2(D4⋊S3), (C3×D16).1C2, C8.10(C3⋊D4), C2.5(C3⋊D16), SmallGroup(192,79)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C48 — D16.S3
 Chief series C1 — C3 — C6 — C12 — C24 — C48 — Dic24 — D16.S3
 Lower central C3 — C6 — C12 — C24 — C48 — D16.S3
 Upper central C1 — C2 — C4 — C8 — C16 — D16

Generators and relations for D16.S3
G = < a,b,c,d | a16=b2=c3=1, d2=a8, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a13b, dcd-1=c-1 >

Character table of D16.S3

 class 1 2A 2B 3 4A 4B 6A 6B 6C 8A 8B 12 16A 16B 16C 16D 24A 24B 32A 32B 32C 32D 32E 32F 32G 32H 48A 48B 48C 48D size 1 1 16 2 2 48 2 16 16 2 2 4 2 2 2 2 4 4 6 6 6 6 6 6 6 6 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ3 1 1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ4 1 1 -1 1 1 -1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 2 2 -2 -1 2 0 -1 1 1 2 2 -1 2 2 2 2 -1 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from D6 ρ6 2 2 0 2 2 0 2 0 0 2 2 2 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 -2 -2 -2 -2 orthogonal lifted from D4 ρ7 2 2 2 -1 2 0 -1 -1 -1 2 2 -1 2 2 2 2 -1 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 2 0 2 -2 0 2 0 0 0 0 -2 -√2 -√2 √2 √2 0 0 -ζ165+ζ163 -ζ1615+ζ169 ζ1615-ζ169 ζ1615-ζ169 ζ165-ζ163 ζ165-ζ163 -ζ165+ζ163 -ζ1615+ζ169 √2 -√2 √2 -√2 orthogonal lifted from D16 ρ9 2 2 0 2 -2 0 2 0 0 0 0 -2 √2 √2 -√2 -√2 0 0 -ζ1615+ζ169 ζ165-ζ163 -ζ165+ζ163 -ζ165+ζ163 ζ1615-ζ169 ζ1615-ζ169 -ζ1615+ζ169 ζ165-ζ163 -√2 √2 -√2 √2 orthogonal lifted from D16 ρ10 2 2 0 2 2 0 2 0 0 -2 -2 2 0 0 0 0 -2 -2 -√2 √2 √2 √2 -√2 -√2 -√2 √2 0 0 0 0 orthogonal lifted from D8 ρ11 2 2 0 2 2 0 2 0 0 -2 -2 2 0 0 0 0 -2 -2 √2 -√2 -√2 -√2 √2 √2 √2 -√2 0 0 0 0 orthogonal lifted from D8 ρ12 2 2 0 2 -2 0 2 0 0 0 0 -2 -√2 -√2 √2 √2 0 0 ζ165-ζ163 ζ1615-ζ169 -ζ1615+ζ169 -ζ1615+ζ169 -ζ165+ζ163 -ζ165+ζ163 ζ165-ζ163 ζ1615-ζ169 √2 -√2 √2 -√2 orthogonal lifted from D16 ρ13 2 2 0 2 -2 0 2 0 0 0 0 -2 √2 √2 -√2 -√2 0 0 ζ1615-ζ169 -ζ165+ζ163 ζ165-ζ163 ζ165-ζ163 -ζ1615+ζ169 -ζ1615+ζ169 ζ1615-ζ169 -ζ165+ζ163 -√2 √2 -√2 √2 orthogonal lifted from D16 ρ14 2 2 0 -1 2 0 -1 -√-3 √-3 2 2 -1 -2 -2 -2 -2 -1 -1 0 0 0 0 0 0 0 0 1 1 1 1 complex lifted from C3⋊D4 ρ15 2 2 0 -1 2 0 -1 √-3 -√-3 2 2 -1 -2 -2 -2 -2 -1 -1 0 0 0 0 0 0 0 0 1 1 1 1 complex lifted from C3⋊D4 ρ16 2 -2 0 2 0 0 -2 0 0 √2 -√2 0 ζ3214-ζ322 -ζ3214+ζ322 ζ3210-ζ326 -ζ3210+ζ326 -√2 √2 ζ3231+ζ3217 ζ3229+ζ3219 ζ3227+ζ3221 ζ3211+ζ325 ζ3225+ζ3223 ζ329+ζ327 ζ3215+ζ32 ζ3213+ζ323 ζ3210-ζ326 ζ3214-ζ322 -ζ3210+ζ326 -ζ3214+ζ322 complex lifted from SD64 ρ17 2 -2 0 2 0 0 -2 0 0 √2 -√2 0 ζ3214-ζ322 -ζ3214+ζ322 ζ3210-ζ326 -ζ3210+ζ326 -√2 √2 ζ3215+ζ32 ζ3213+ζ323 ζ3211+ζ325 ζ3227+ζ3221 ζ329+ζ327 ζ3225+ζ3223 ζ3231+ζ3217 ζ3229+ζ3219 ζ3210-ζ326 ζ3214-ζ322 -ζ3210+ζ326 -ζ3214+ζ322 complex lifted from SD64 ρ18 2 -2 0 2 0 0 -2 0 0 -√2 √2 0 -ζ3210+ζ326 ζ3210-ζ326 ζ3214-ζ322 -ζ3214+ζ322 √2 -√2 ζ3211+ζ325 ζ3215+ζ32 ζ3225+ζ3223 ζ329+ζ327 ζ3213+ζ323 ζ3229+ζ3219 ζ3227+ζ3221 ζ3231+ζ3217 ζ3214-ζ322 -ζ3210+ζ326 -ζ3214+ζ322 ζ3210-ζ326 complex lifted from SD64 ρ19 2 -2 0 2 0 0 -2 0 0 √2 -√2 0 -ζ3214+ζ322 ζ3214-ζ322 -ζ3210+ζ326 ζ3210-ζ326 -√2 √2 ζ329+ζ327 ζ3227+ζ3221 ζ3213+ζ323 ζ3229+ζ3219 ζ3231+ζ3217 ζ3215+ζ32 ζ3225+ζ3223 ζ3211+ζ325 -ζ3210+ζ326 -ζ3214+ζ322 ζ3210-ζ326 ζ3214-ζ322 complex lifted from SD64 ρ20 2 -2 0 2 0 0 -2 0 0 -√2 √2 0 ζ3210-ζ326 -ζ3210+ζ326 -ζ3214+ζ322 ζ3214-ζ322 √2 -√2 ζ3213+ζ323 ζ329+ζ327 ζ3215+ζ32 ζ3231+ζ3217 ζ3227+ζ3221 ζ3211+ζ325 ζ3229+ζ3219 ζ3225+ζ3223 -ζ3214+ζ322 ζ3210-ζ326 ζ3214-ζ322 -ζ3210+ζ326 complex lifted from SD64 ρ21 2 -2 0 2 0 0 -2 0 0 -√2 √2 0 ζ3210-ζ326 -ζ3210+ζ326 -ζ3214+ζ322 ζ3214-ζ322 √2 -√2 ζ3229+ζ3219 ζ3225+ζ3223 ζ3231+ζ3217 ζ3215+ζ32 ζ3211+ζ325 ζ3227+ζ3221 ζ3213+ζ323 ζ329+ζ327 -ζ3214+ζ322 ζ3210-ζ326 ζ3214-ζ322 -ζ3210+ζ326 complex lifted from SD64 ρ22 2 -2 0 2 0 0 -2 0 0 √2 -√2 0 -ζ3214+ζ322 ζ3214-ζ322 -ζ3210+ζ326 ζ3210-ζ326 -√2 √2 ζ3225+ζ3223 ζ3211+ζ325 ζ3229+ζ3219 ζ3213+ζ323 ζ3215+ζ32 ζ3231+ζ3217 ζ329+ζ327 ζ3227+ζ3221 -ζ3210+ζ326 -ζ3214+ζ322 ζ3210-ζ326 ζ3214-ζ322 complex lifted from SD64 ρ23 2 -2 0 2 0 0 -2 0 0 -√2 √2 0 -ζ3210+ζ326 ζ3210-ζ326 ζ3214-ζ322 -ζ3214+ζ322 √2 -√2 ζ3227+ζ3221 ζ3231+ζ3217 ζ329+ζ327 ζ3225+ζ3223 ζ3229+ζ3219 ζ3213+ζ323 ζ3211+ζ325 ζ3215+ζ32 ζ3214-ζ322 -ζ3210+ζ326 -ζ3214+ζ322 ζ3210-ζ326 complex lifted from SD64 ρ24 4 4 0 -2 4 0 -2 0 0 -4 -4 -2 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ25 4 4 0 -2 -4 0 -2 0 0 0 0 2 -2√2 -2√2 2√2 2√2 0 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 orthogonal lifted from C3⋊D16, Schur index 2 ρ26 4 4 0 -2 -4 0 -2 0 0 0 0 2 2√2 2√2 -2√2 -2√2 0 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 orthogonal lifted from C3⋊D16, Schur index 2 ρ27 4 -4 0 -2 0 0 2 0 0 2√2 -2√2 0 -2ζ167+2ζ16 2ζ167-2ζ16 -2ζ165+2ζ163 2ζ165-2ζ163 √2 -√2 0 0 0 0 0 0 0 0 ζ165-ζ163 -ζ1615+ζ169 -ζ165+ζ163 ζ1615-ζ169 symplectic faithful, Schur index 2 ρ28 4 -4 0 -2 0 0 2 0 0 -2√2 2√2 0 2ζ165-2ζ163 -2ζ165+2ζ163 -2ζ167+2ζ16 2ζ167-2ζ16 -√2 √2 0 0 0 0 0 0 0 0 -ζ1615+ζ169 -ζ165+ζ163 ζ1615-ζ169 ζ165-ζ163 symplectic faithful, Schur index 2 ρ29 4 -4 0 -2 0 0 2 0 0 2√2 -2√2 0 2ζ167-2ζ16 -2ζ167+2ζ16 2ζ165-2ζ163 -2ζ165+2ζ163 √2 -√2 0 0 0 0 0 0 0 0 -ζ165+ζ163 ζ1615-ζ169 ζ165-ζ163 -ζ1615+ζ169 symplectic faithful, Schur index 2 ρ30 4 -4 0 -2 0 0 2 0 0 -2√2 2√2 0 -2ζ165+2ζ163 2ζ165-2ζ163 2ζ167-2ζ16 -2ζ167+2ζ16 -√2 √2 0 0 0 0 0 0 0 0 ζ1615-ζ169 ζ165-ζ163 -ζ1615+ζ169 -ζ165+ζ163 symplectic faithful, Schur index 2

Smallest permutation representation of D16.S3
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)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 27)(18 26)(19 25)(20 24)(21 23)(28 32)(29 31)(33 42)(34 41)(35 40)(36 39)(37 38)(43 48)(44 47)(45 46)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(78 80)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(93 96)(94 95)
(1 38 95)(2 39 96)(3 40 81)(4 41 82)(5 42 83)(6 43 84)(7 44 85)(8 45 86)(9 46 87)(10 47 88)(11 48 89)(12 33 90)(13 34 91)(14 35 92)(15 36 93)(16 37 94)(17 59 66)(18 60 67)(19 61 68)(20 62 69)(21 63 70)(22 64 71)(23 49 72)(24 50 73)(25 51 74)(26 52 75)(27 53 76)(28 54 77)(29 55 78)(30 56 79)(31 57 80)(32 58 65)
(1 31 9 23)(2 30 10 22)(3 29 11 21)(4 28 12 20)(5 27 13 19)(6 26 14 18)(7 25 15 17)(8 24 16 32)(33 69 41 77)(34 68 42 76)(35 67 43 75)(36 66 44 74)(37 65 45 73)(38 80 46 72)(39 79 47 71)(40 78 48 70)(49 95 57 87)(50 94 58 86)(51 93 59 85)(52 92 60 84)(53 91 61 83)(54 90 62 82)(55 89 63 81)(56 88 64 96)```

`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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,27)(18,26)(19,25)(20,24)(21,23)(28,32)(29,31)(33,42)(34,41)(35,40)(36,39)(37,38)(43,48)(44,47)(45,46)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(78,80)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(93,96)(94,95), (1,38,95)(2,39,96)(3,40,81)(4,41,82)(5,42,83)(6,43,84)(7,44,85)(8,45,86)(9,46,87)(10,47,88)(11,48,89)(12,33,90)(13,34,91)(14,35,92)(15,36,93)(16,37,94)(17,59,66)(18,60,67)(19,61,68)(20,62,69)(21,63,70)(22,64,71)(23,49,72)(24,50,73)(25,51,74)(26,52,75)(27,53,76)(28,54,77)(29,55,78)(30,56,79)(31,57,80)(32,58,65), (1,31,9,23)(2,30,10,22)(3,29,11,21)(4,28,12,20)(5,27,13,19)(6,26,14,18)(7,25,15,17)(8,24,16,32)(33,69,41,77)(34,68,42,76)(35,67,43,75)(36,66,44,74)(37,65,45,73)(38,80,46,72)(39,79,47,71)(40,78,48,70)(49,95,57,87)(50,94,58,86)(51,93,59,85)(52,92,60,84)(53,91,61,83)(54,90,62,82)(55,89,63,81)(56,88,64,96)>;`

`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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,27)(18,26)(19,25)(20,24)(21,23)(28,32)(29,31)(33,42)(34,41)(35,40)(36,39)(37,38)(43,48)(44,47)(45,46)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(78,80)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(93,96)(94,95), (1,38,95)(2,39,96)(3,40,81)(4,41,82)(5,42,83)(6,43,84)(7,44,85)(8,45,86)(9,46,87)(10,47,88)(11,48,89)(12,33,90)(13,34,91)(14,35,92)(15,36,93)(16,37,94)(17,59,66)(18,60,67)(19,61,68)(20,62,69)(21,63,70)(22,64,71)(23,49,72)(24,50,73)(25,51,74)(26,52,75)(27,53,76)(28,54,77)(29,55,78)(30,56,79)(31,57,80)(32,58,65), (1,31,9,23)(2,30,10,22)(3,29,11,21)(4,28,12,20)(5,27,13,19)(6,26,14,18)(7,25,15,17)(8,24,16,32)(33,69,41,77)(34,68,42,76)(35,67,43,75)(36,66,44,74)(37,65,45,73)(38,80,46,72)(39,79,47,71)(40,78,48,70)(49,95,57,87)(50,94,58,86)(51,93,59,85)(52,92,60,84)(53,91,61,83)(54,90,62,82)(55,89,63,81)(56,88,64,96) );`

`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)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,27),(18,26),(19,25),(20,24),(21,23),(28,32),(29,31),(33,42),(34,41),(35,40),(36,39),(37,38),(43,48),(44,47),(45,46),(49,63),(50,62),(51,61),(52,60),(53,59),(54,58),(55,57),(65,77),(66,76),(67,75),(68,74),(69,73),(70,72),(78,80),(81,92),(82,91),(83,90),(84,89),(85,88),(86,87),(93,96),(94,95)], [(1,38,95),(2,39,96),(3,40,81),(4,41,82),(5,42,83),(6,43,84),(7,44,85),(8,45,86),(9,46,87),(10,47,88),(11,48,89),(12,33,90),(13,34,91),(14,35,92),(15,36,93),(16,37,94),(17,59,66),(18,60,67),(19,61,68),(20,62,69),(21,63,70),(22,64,71),(23,49,72),(24,50,73),(25,51,74),(26,52,75),(27,53,76),(28,54,77),(29,55,78),(30,56,79),(31,57,80),(32,58,65)], [(1,31,9,23),(2,30,10,22),(3,29,11,21),(4,28,12,20),(5,27,13,19),(6,26,14,18),(7,25,15,17),(8,24,16,32),(33,69,41,77),(34,68,42,76),(35,67,43,75),(36,66,44,74),(37,65,45,73),(38,80,46,72),(39,79,47,71),(40,78,48,70),(49,95,57,87),(50,94,58,86),(51,93,59,85),(52,92,60,84),(53,91,61,83),(54,90,62,82),(55,89,63,81),(56,88,64,96)]])`

Matrix representation of D16.S3 in GL4(𝔽97) generated by

 2 26 0 0 71 2 0 0 0 0 96 0 0 0 0 96
,
 2 26 0 0 26 95 0 0 0 0 96 19 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 35 53 0 0 0 61
,
 85 7 0 0 7 12 0 0 0 0 19 14 0 0 2 78
`G:=sub<GL(4,GF(97))| [2,71,0,0,26,2,0,0,0,0,96,0,0,0,0,96],[2,26,0,0,26,95,0,0,0,0,96,0,0,0,19,1],[1,0,0,0,0,1,0,0,0,0,35,0,0,0,53,61],[85,7,0,0,7,12,0,0,0,0,19,2,0,0,14,78] >;`

D16.S3 in GAP, Magma, Sage, TeX

`D_{16}.S_3`
`% in TeX`

`G:=Group("D16.S3");`
`// GroupNames label`

`G:=SmallGroup(192,79);`
`// by ID`

`G=gap.SmallGroup(192,79);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,85,254,135,142,675,346,192,1684,851,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^16=b^2=c^3=1,d^2=a^8,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^13*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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