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## G = Q8⋊D12order 192 = 26·3

### The semidirect product of Q8 and D12 acting via D12/C4=S3

Aliases: Q8⋊D12, C4⋊GL2(𝔽3), SL2(𝔽3)⋊1D4, (C4×Q8)⋊2S3, C2.4(C4⋊S4), (C2×C4).11S4, (C2×Q8).9D6, C22.36(C2×S4), C2.4(C4.3S4), (C4×SL2(𝔽3))⋊6C2, (C2×GL2(𝔽3))⋊4C2, C2.4(C2×GL2(𝔽3)), (C2×SL2(𝔽3)).9C22, SmallGroup(192,952)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C2×SL2(𝔽3) — Q8⋊D12
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — C2×GL2(𝔽3) — Q8⋊D12
 Lower central SL2(𝔽3) — C2×SL2(𝔽3) — Q8⋊D12
 Upper central C1 — C22 — C2×C4

Generators and relations for Q8⋊D12
G = < a,b,c,d | a4=c12=d2=1, b2=a2, bab-1=dad=a-1, cac-1=ab, cbc-1=a, dbd=a-1b, dcd=c-1 >

Subgroups: 473 in 91 conjugacy classes, 17 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C12, D6, C2×C6, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×Q8, SL2(𝔽3), D12, C2×C12, C22×S3, D4⋊C4, C4⋊C8, C4×Q8, C41D4, C2×SD16, GL2(𝔽3), C2×SL2(𝔽3), C2×D12, C4⋊SD16, C4×SL2(𝔽3), C2×GL2(𝔽3), Q8⋊D12
Quotients: C1, C2, C22, S3, D4, D6, D12, S4, GL2(𝔽3), C2×S4, C4⋊S4, C2×GL2(𝔽3), C4.3S4, Q8⋊D12

Character table of Q8⋊D12

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D size 1 1 1 1 24 24 8 2 2 6 6 12 8 8 8 12 12 12 12 8 8 8 8 ρ1 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 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 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 linear of order 2 ρ5 2 2 2 2 0 0 -1 -2 -2 2 2 -2 -1 -1 -1 0 0 0 0 1 1 1 1 orthogonal lifted from D6 ρ6 2 2 -2 -2 0 0 2 0 0 -2 2 0 -2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ7 2 2 2 2 0 0 -1 2 2 2 2 2 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 2 -2 -2 0 0 -1 0 0 -2 2 0 1 -1 1 0 0 0 0 √3 √3 -√3 -√3 orthogonal lifted from D12 ρ9 2 2 -2 -2 0 0 -1 0 0 -2 2 0 1 -1 1 0 0 0 0 -√3 -√3 √3 √3 orthogonal lifted from D12 ρ10 2 -2 2 -2 0 0 -1 2 -2 0 0 0 1 1 -1 √-2 √-2 -√-2 -√-2 1 -1 1 -1 complex lifted from GL2(𝔽3) ρ11 2 -2 2 -2 0 0 -1 -2 2 0 0 0 1 1 -1 -√-2 √-2 √-2 -√-2 -1 1 -1 1 complex lifted from GL2(𝔽3) ρ12 2 -2 2 -2 0 0 -1 2 -2 0 0 0 1 1 -1 -√-2 -√-2 √-2 √-2 1 -1 1 -1 complex lifted from GL2(𝔽3) ρ13 2 -2 2 -2 0 0 -1 -2 2 0 0 0 1 1 -1 √-2 -√-2 -√-2 √-2 -1 1 -1 1 complex lifted from GL2(𝔽3) ρ14 3 3 3 3 -1 1 0 -3 -3 -1 -1 1 0 0 0 1 -1 1 -1 0 0 0 0 orthogonal lifted from C2×S4 ρ15 3 3 3 3 1 -1 0 -3 -3 -1 -1 1 0 0 0 -1 1 -1 1 0 0 0 0 orthogonal lifted from C2×S4 ρ16 3 3 3 3 -1 -1 0 3 3 -1 -1 -1 0 0 0 1 1 1 1 0 0 0 0 orthogonal lifted from S4 ρ17 3 3 3 3 1 1 0 3 3 -1 -1 -1 0 0 0 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from S4 ρ18 4 -4 4 -4 0 0 1 4 -4 0 0 0 -1 -1 1 0 0 0 0 -1 1 -1 1 orthogonal lifted from GL2(𝔽3) ρ19 4 -4 -4 4 0 0 -2 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from C4.3S4 ρ20 4 -4 4 -4 0 0 1 -4 4 0 0 0 -1 -1 1 0 0 0 0 1 -1 1 -1 orthogonal lifted from GL2(𝔽3) ρ21 4 -4 -4 4 0 0 1 0 0 0 0 0 1 -1 -1 0 0 0 0 √3 -√3 -√3 √3 orthogonal lifted from C4.3S4 ρ22 4 -4 -4 4 0 0 1 0 0 0 0 0 1 -1 -1 0 0 0 0 -√3 √3 √3 -√3 orthogonal lifted from C4.3S4 ρ23 6 6 -6 -6 0 0 0 0 0 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4⋊S4

Smallest permutation representation of Q8⋊D12
On 32 points
Generators in S32
```(1 13 5 22)(2 10 6 31)(3 19 7 28)(4 16 8 25)(9 17 30 26)(11 24 32 15)(12 20 21 29)(14 27 23 18)
(1 17 5 26)(2 14 6 23)(3 11 7 32)(4 20 8 29)(9 22 30 13)(10 18 31 27)(12 25 21 16)(15 28 24 19)
(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)
(1 3)(5 7)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)```

`G:=sub<Sym(32)| (1,13,5,22)(2,10,6,31)(3,19,7,28)(4,16,8,25)(9,17,30,26)(11,24,32,15)(12,20,21,29)(14,27,23,18), (1,17,5,26)(2,14,6,23)(3,11,7,32)(4,20,8,29)(9,22,30,13)(10,18,31,27)(12,25,21,16)(15,28,24,19), (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), (1,3)(5,7)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)>;`

`G:=Group( (1,13,5,22)(2,10,6,31)(3,19,7,28)(4,16,8,25)(9,17,30,26)(11,24,32,15)(12,20,21,29)(14,27,23,18), (1,17,5,26)(2,14,6,23)(3,11,7,32)(4,20,8,29)(9,22,30,13)(10,18,31,27)(12,25,21,16)(15,28,24,19), (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), (1,3)(5,7)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21) );`

`G=PermutationGroup([[(1,13,5,22),(2,10,6,31),(3,19,7,28),(4,16,8,25),(9,17,30,26),(11,24,32,15),(12,20,21,29),(14,27,23,18)], [(1,17,5,26),(2,14,6,23),(3,11,7,32),(4,20,8,29),(9,22,30,13),(10,18,31,27),(12,25,21,16),(15,28,24,19)], [(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)], [(1,3),(5,7),(9,32),(10,31),(11,30),(12,29),(13,28),(14,27),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21)]])`

Matrix representation of Q8⋊D12 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 41 21 0 0 52 32
,
 1 0 0 0 0 1 0 0 0 0 53 21 0 0 40 20
,
 72 2 0 0 72 1 0 0 0 0 0 1 0 0 72 72
,
 72 0 0 0 72 1 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,41,52,0,0,21,32],[1,0,0,0,0,1,0,0,0,0,53,40,0,0,21,20],[72,72,0,0,2,1,0,0,0,0,0,72,0,0,1,72],[72,72,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;`

Q8⋊D12 in GAP, Magma, Sage, TeX

`Q_8\rtimes D_{12}`
`% in TeX`

`G:=Group("Q8:D12");`
`// GroupNames label`

`G:=SmallGroup(192,952);`
`// by ID`

`G=gap.SmallGroup(192,952);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,85,36,451,1684,655,172,1013,404,285,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=c^12=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,c*a*c^-1=a*b,c*b*c^-1=a,d*b*d=a^-1*b,d*c*d=c^-1>;`
`// generators/relations`

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