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G = Q16⋊D6order 192 = 26·3

2nd semidirect product of Q16 and D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — Q16⋊D6
 Chief series C1 — C3 — C6 — C12 — C24 — D24 — C2×D24 — Q16⋊D6
 Lower central C3 — C6 — C12 — C24 — Q16⋊D6
 Upper central C1 — C2 — C2×C4 — C2×C8 — C4○D8

Generators and relations for Q16⋊D6
G = < a,b,c,d | a8=c6=d2=1, b2=a4, bab-1=dad=a-1, ac=ca, cbc-1=a4b, dbd=a3b, dcd=c-1 >

Subgroups: 360 in 90 conjugacy classes, 35 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C12, C12, D6, C2×C6, C2×C6, C16, C2×C8, D8, D8, SD16, Q16, C2×D4, C4○D4, C24, D12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, M5(2), D16, SD32, C2×D8, C4○D8, C3⋊C16, D24, D24, C2×C24, C3×D8, C3×SD16, C3×Q16, C2×D12, C3×C4○D4, C16⋊C22, C12.C8, C3⋊D16, C8.6D6, C2×D24, C3×C4○D8, Q16⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C3⋊D4, C22×S3, C2×D8, D4⋊S3, C2×C3⋊D4, C16⋊C22, C2×D4⋊S3, Q16⋊D6

Character table of Q16⋊D6

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 6A 6B 6C 6D 8A 8B 8C 12A 12B 12C 12D 12E 16A 16B 16C 16D 24A 24B 24C 24D size 1 1 2 8 24 24 2 2 2 8 2 4 8 8 2 2 4 2 2 4 8 8 12 12 12 12 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 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 ρ6 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 ρ7 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 ρ8 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 ρ9 2 2 2 0 0 0 2 2 2 0 2 2 0 0 -2 -2 -2 2 2 2 0 0 0 0 0 0 -2 -2 -2 -2 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 -1 2 2 2 -1 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 2 -2 0 0 -1 2 2 -2 -1 -1 1 1 2 2 2 -1 -1 -1 1 1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from D6 ρ12 2 2 -2 0 0 0 2 -2 2 0 2 -2 0 0 -2 -2 2 -2 -2 2 0 0 0 0 0 0 2 -2 2 -2 orthogonal lifted from D4 ρ13 2 2 -2 2 0 0 -1 -2 2 -2 -1 1 -1 -1 2 2 -2 1 1 -1 1 1 0 0 0 0 1 -1 1 -1 orthogonal lifted from D6 ρ14 2 2 -2 -2 0 0 -1 -2 2 2 -1 1 1 1 2 2 -2 1 1 -1 -1 -1 0 0 0 0 1 -1 1 -1 orthogonal lifted from D6 ρ15 2 2 -2 0 0 0 2 2 -2 0 2 -2 0 0 0 0 0 2 2 -2 0 0 -√2 √2 -√2 √2 0 0 0 0 orthogonal lifted from D8 ρ16 2 2 -2 0 0 0 2 2 -2 0 2 -2 0 0 0 0 0 2 2 -2 0 0 √2 -√2 √2 -√2 0 0 0 0 orthogonal lifted from D8 ρ17 2 2 2 0 0 0 2 -2 -2 0 2 2 0 0 0 0 0 -2 -2 -2 0 0 -√2 -√2 √2 √2 0 0 0 0 orthogonal lifted from D8 ρ18 2 2 2 0 0 0 2 -2 -2 0 2 2 0 0 0 0 0 -2 -2 -2 0 0 √2 √2 -√2 -√2 0 0 0 0 orthogonal lifted from D8 ρ19 2 2 -2 0 0 0 -1 -2 2 0 -1 1 √-3 -√-3 -2 -2 2 1 1 -1 √-3 -√-3 0 0 0 0 -1 1 -1 1 complex lifted from C3⋊D4 ρ20 2 2 2 0 0 0 -1 2 2 0 -1 -1 -√-3 √-3 -2 -2 -2 -1 -1 -1 √-3 -√-3 0 0 0 0 1 1 1 1 complex lifted from C3⋊D4 ρ21 2 2 -2 0 0 0 -1 -2 2 0 -1 1 -√-3 √-3 -2 -2 2 1 1 -1 -√-3 √-3 0 0 0 0 -1 1 -1 1 complex lifted from C3⋊D4 ρ22 2 2 2 0 0 0 -1 2 2 0 -1 -1 √-3 -√-3 -2 -2 -2 -1 -1 -1 -√-3 √-3 0 0 0 0 1 1 1 1 complex lifted from C3⋊D4 ρ23 4 4 4 0 0 0 -2 -4 -4 0 -2 -2 0 0 0 0 0 2 2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ24 4 4 -4 0 0 0 -2 4 -4 0 -2 2 0 0 0 0 0 -2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ25 4 -4 0 0 0 0 4 0 0 0 -4 0 0 0 2√2 -2√2 0 0 0 0 0 0 0 0 0 0 0 -2√2 0 2√2 orthogonal lifted from C16⋊C22 ρ26 4 -4 0 0 0 0 4 0 0 0 -4 0 0 0 -2√2 2√2 0 0 0 0 0 0 0 0 0 0 0 2√2 0 -2√2 orthogonal lifted from C16⋊C22 ρ27 4 -4 0 0 0 0 -2 0 0 0 2 0 0 0 2√2 -2√2 0 -2√3 2√3 0 0 0 0 0 0 0 √6 √2 -√6 -√2 orthogonal faithful ρ28 4 -4 0 0 0 0 -2 0 0 0 2 0 0 0 2√2 -2√2 0 2√3 -2√3 0 0 0 0 0 0 0 -√6 √2 √6 -√2 orthogonal faithful ρ29 4 -4 0 0 0 0 -2 0 0 0 2 0 0 0 -2√2 2√2 0 2√3 -2√3 0 0 0 0 0 0 0 √6 -√2 -√6 √2 orthogonal faithful ρ30 4 -4 0 0 0 0 -2 0 0 0 2 0 0 0 -2√2 2√2 0 -2√3 2√3 0 0 0 0 0 0 0 -√6 -√2 √6 √2 orthogonal faithful

Smallest permutation representation of Q16⋊D6
On 48 points
Generators in S48
```(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)
(1 13 5 9)(2 12 6 16)(3 11 7 15)(4 10 8 14)(17 28 21 32)(18 27 22 31)(19 26 23 30)(20 25 24 29)(33 47 37 43)(34 46 38 42)(35 45 39 41)(36 44 40 48)
(1 38 25)(2 39 26)(3 40 27)(4 33 28)(5 34 29)(6 35 30)(7 36 31)(8 37 32)(9 42 20 13 46 24)(10 43 21 14 47 17)(11 44 22 15 48 18)(12 45 23 16 41 19)
(1 25)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 17)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(33 35)(36 40)(37 39)(41 48)(42 47)(43 46)(44 45)```

`G:=sub<Sym(48)| (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), (1,13,5,9)(2,12,6,16)(3,11,7,15)(4,10,8,14)(17,28,21,32)(18,27,22,31)(19,26,23,30)(20,25,24,29)(33,47,37,43)(34,46,38,42)(35,45,39,41)(36,44,40,48), (1,38,25)(2,39,26)(3,40,27)(4,33,28)(5,34,29)(6,35,30)(7,36,31)(8,37,32)(9,42,20,13,46,24)(10,43,21,14,47,17)(11,44,22,15,48,18)(12,45,23,16,41,19), (1,25)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,17)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(33,35)(36,40)(37,39)(41,48)(42,47)(43,46)(44,45)>;`

`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), (1,13,5,9)(2,12,6,16)(3,11,7,15)(4,10,8,14)(17,28,21,32)(18,27,22,31)(19,26,23,30)(20,25,24,29)(33,47,37,43)(34,46,38,42)(35,45,39,41)(36,44,40,48), (1,38,25)(2,39,26)(3,40,27)(4,33,28)(5,34,29)(6,35,30)(7,36,31)(8,37,32)(9,42,20,13,46,24)(10,43,21,14,47,17)(11,44,22,15,48,18)(12,45,23,16,41,19), (1,25)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,17)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(33,35)(36,40)(37,39)(41,48)(42,47)(43,46)(44,45) );`

`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)], [(1,13,5,9),(2,12,6,16),(3,11,7,15),(4,10,8,14),(17,28,21,32),(18,27,22,31),(19,26,23,30),(20,25,24,29),(33,47,37,43),(34,46,38,42),(35,45,39,41),(36,44,40,48)], [(1,38,25),(2,39,26),(3,40,27),(4,33,28),(5,34,29),(6,35,30),(7,36,31),(8,37,32),(9,42,20,13,46,24),(10,43,21,14,47,17),(11,44,22,15,48,18),(12,45,23,16,41,19)], [(1,25),(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,17),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18),(33,35),(36,40),(37,39),(41,48),(42,47),(43,46),(44,45)]])`

Matrix representation of Q16⋊D6 in GL4(𝔽97) generated by

 16 18 0 0 79 95 0 0 0 0 95 79 0 0 18 16
,
 0 0 96 0 0 0 0 96 1 0 0 0 0 1 0 0
,
 96 96 0 0 1 0 0 0 0 0 1 1 0 0 96 0
,
 96 96 0 0 0 1 0 0 0 0 95 16 0 0 18 2
`G:=sub<GL(4,GF(97))| [16,79,0,0,18,95,0,0,0,0,95,18,0,0,79,16],[0,0,1,0,0,0,0,1,96,0,0,0,0,96,0,0],[96,1,0,0,96,0,0,0,0,0,1,96,0,0,1,0],[96,0,0,0,96,1,0,0,0,0,95,18,0,0,16,2] >;`

Q16⋊D6 in GAP, Magma, Sage, TeX

`Q_{16}\rtimes D_6`
`% in TeX`

`G:=Group("Q16:D6");`
`// GroupNames label`

`G:=SmallGroup(192,752);`
`// by ID`

`G=gap.SmallGroup(192,752);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,387,675,185,192,1684,438,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=c^6=d^2=1,b^2=a^4,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^4*b,d*b*d=a^3*b,d*c*d=c^-1>;`
`// generators/relations`

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