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## G = Q8.D12order 192 = 26·3

### 1st non-split extension by Q8 of D12 acting via D12/C4=S3

Aliases: Q8.1D12, C4⋊CSU2(𝔽3), SL2(𝔽3).1D4, (C2×C4).9S4, C2.3(C4⋊S4), (C2×Q8).6D6, (C4×Q8).6S3, C22.34(C2×S4), C2.4(C4.S4), (C4×SL2(𝔽3)).5C2, C2.3(C2×CSU2(𝔽3)), (C2×CSU2(𝔽3)).1C2, (C2×SL2(𝔽3)).6C22, SmallGroup(192,949)

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×CSU2(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=1, b2=d2=a2, bab-1=dad-1=a-1, cac-1=ab, cbc-1=a, dbd-1=a-1b, dcd-1=c-1 >

Subgroups: 249 in 67 conjugacy classes, 17 normal (13 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Q8, Dic3, C12, C2×C6, C42, C4⋊C4, C2×C8, Q16, C2×Q8, C2×Q8, SL2(𝔽3), C2×Dic3, C2×C12, Q8⋊C4, C4⋊C8, C4×Q8, C4⋊Q8, C2×Q16, C4⋊Dic3, CSU2(𝔽3), C2×SL2(𝔽3), C42Q16, C4×SL2(𝔽3), C2×CSU2(𝔽3), Q8.D12
Quotients: C1, C2, C22, S3, D4, D6, D12, S4, CSU2(𝔽3), C2×S4, C4⋊S4, C2×CSU2(𝔽3), C4.S4, Q8.D12

Character table of Q8.D12

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D size 1 1 1 1 8 2 2 6 6 12 24 24 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 2 0 0 -2 2 0 0 0 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ6 2 2 2 2 -1 -2 -2 2 2 -2 0 0 -1 -1 -1 0 0 0 0 1 1 1 1 orthogonal lifted from D6 ρ7 2 2 2 2 -1 2 2 2 2 2 0 0 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 2 -2 -2 -1 0 0 -2 2 0 0 0 1 1 -1 0 0 0 0 √3 √3 -√3 -√3 orthogonal lifted from D12 ρ9 2 2 -2 -2 -1 0 0 -2 2 0 0 0 1 1 -1 0 0 0 0 -√3 -√3 √3 √3 orthogonal lifted from D12 ρ10 2 -2 2 -2 -1 -2 2 0 0 0 0 0 1 -1 1 √2 -√2 -√2 √2 -1 1 -1 1 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ11 2 -2 2 -2 -1 2 -2 0 0 0 0 0 1 -1 1 √2 √2 -√2 -√2 1 -1 1 -1 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ12 2 -2 2 -2 -1 -2 2 0 0 0 0 0 1 -1 1 -√2 √2 √2 -√2 -1 1 -1 1 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ13 2 -2 2 -2 -1 2 -2 0 0 0 0 0 1 -1 1 -√2 -√2 √2 √2 1 -1 1 -1 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ14 3 3 3 3 0 3 3 -1 -1 -1 -1 -1 0 0 0 1 1 1 1 0 0 0 0 orthogonal lifted from S4 ρ15 3 3 3 3 0 -3 -3 -1 -1 1 -1 1 0 0 0 1 -1 1 -1 0 0 0 0 orthogonal lifted from C2×S4 ρ16 3 3 3 3 0 -3 -3 -1 -1 1 1 -1 0 0 0 -1 1 -1 1 0 0 0 0 orthogonal lifted from C2×S4 ρ17 3 3 3 3 0 3 3 -1 -1 -1 1 1 0 0 0 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from S4 ρ18 4 -4 4 -4 1 4 -4 0 0 0 0 0 -1 1 -1 0 0 0 0 -1 1 -1 1 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ19 4 -4 4 -4 1 -4 4 0 0 0 0 0 -1 1 -1 0 0 0 0 1 -1 1 -1 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ20 4 -4 -4 4 -2 0 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 0 0 symplectic lifted from C4.S4, Schur index 2 ρ21 4 -4 -4 4 1 0 0 0 0 0 0 0 1 -1 -1 0 0 0 0 √3 -√3 -√3 √3 symplectic lifted from C4.S4, Schur index 2 ρ22 4 -4 -4 4 1 0 0 0 0 0 0 0 1 -1 -1 0 0 0 0 -√3 √3 √3 -√3 symplectic lifted from C4.S4, Schur index 2 ρ23 6 6 -6 -6 0 0 0 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4⋊S4

Smallest permutation representation of Q8.D12
On 64 points
Generators in S64
```(1 30 11 59)(2 39 12 56)(3 36 9 53)(4 33 10 62)(5 49 16 24)(6 46 13 21)(7 43 14 18)(8 52 15 27)(17 25 42 50)(19 48 44 23)(20 28 45 41)(22 51 47 26)(29 37 58 54)(31 64 60 35)(32 40 61 57)(34 55 63 38)
(1 34 11 63)(2 31 12 60)(3 40 9 57)(4 37 10 54)(5 41 16 28)(6 50 13 25)(7 47 14 22)(8 44 15 19)(17 46 42 21)(18 26 43 51)(20 49 45 24)(23 52 48 27)(29 62 58 33)(30 38 59 55)(32 53 61 36)(35 56 64 39)
(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)
(1 8 11 15)(2 7 12 14)(3 6 9 13)(4 5 10 16)(17 57 42 40)(18 56 43 39)(19 55 44 38)(20 54 45 37)(21 53 46 36)(22 64 47 35)(23 63 48 34)(24 62 49 33)(25 61 50 32)(26 60 51 31)(27 59 52 30)(28 58 41 29)```

`G:=sub<Sym(64)| (1,30,11,59)(2,39,12,56)(3,36,9,53)(4,33,10,62)(5,49,16,24)(6,46,13,21)(7,43,14,18)(8,52,15,27)(17,25,42,50)(19,48,44,23)(20,28,45,41)(22,51,47,26)(29,37,58,54)(31,64,60,35)(32,40,61,57)(34,55,63,38), (1,34,11,63)(2,31,12,60)(3,40,9,57)(4,37,10,54)(5,41,16,28)(6,50,13,25)(7,47,14,22)(8,44,15,19)(17,46,42,21)(18,26,43,51)(20,49,45,24)(23,52,48,27)(29,62,58,33)(30,38,59,55)(32,53,61,36)(35,56,64,39), (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), (1,8,11,15)(2,7,12,14)(3,6,9,13)(4,5,10,16)(17,57,42,40)(18,56,43,39)(19,55,44,38)(20,54,45,37)(21,53,46,36)(22,64,47,35)(23,63,48,34)(24,62,49,33)(25,61,50,32)(26,60,51,31)(27,59,52,30)(28,58,41,29)>;`

`G:=Group( (1,30,11,59)(2,39,12,56)(3,36,9,53)(4,33,10,62)(5,49,16,24)(6,46,13,21)(7,43,14,18)(8,52,15,27)(17,25,42,50)(19,48,44,23)(20,28,45,41)(22,51,47,26)(29,37,58,54)(31,64,60,35)(32,40,61,57)(34,55,63,38), (1,34,11,63)(2,31,12,60)(3,40,9,57)(4,37,10,54)(5,41,16,28)(6,50,13,25)(7,47,14,22)(8,44,15,19)(17,46,42,21)(18,26,43,51)(20,49,45,24)(23,52,48,27)(29,62,58,33)(30,38,59,55)(32,53,61,36)(35,56,64,39), (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), (1,8,11,15)(2,7,12,14)(3,6,9,13)(4,5,10,16)(17,57,42,40)(18,56,43,39)(19,55,44,38)(20,54,45,37)(21,53,46,36)(22,64,47,35)(23,63,48,34)(24,62,49,33)(25,61,50,32)(26,60,51,31)(27,59,52,30)(28,58,41,29) );`

`G=PermutationGroup([[(1,30,11,59),(2,39,12,56),(3,36,9,53),(4,33,10,62),(5,49,16,24),(6,46,13,21),(7,43,14,18),(8,52,15,27),(17,25,42,50),(19,48,44,23),(20,28,45,41),(22,51,47,26),(29,37,58,54),(31,64,60,35),(32,40,61,57),(34,55,63,38)], [(1,34,11,63),(2,31,12,60),(3,40,9,57),(4,37,10,54),(5,41,16,28),(6,50,13,25),(7,47,14,22),(8,44,15,19),(17,46,42,21),(18,26,43,51),(20,49,45,24),(23,52,48,27),(29,62,58,33),(30,38,59,55),(32,53,61,36),(35,56,64,39)], [(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)], [(1,8,11,15),(2,7,12,14),(3,6,9,13),(4,5,10,16),(17,57,42,40),(18,56,43,39),(19,55,44,38),(20,54,45,37),(21,53,46,36),(22,64,47,35),(23,63,48,34),(24,62,49,33),(25,61,50,32),(26,60,51,31),(27,59,52,30),(28,58,41,29)]])`

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

 1 0 0 0 0 1 0 0 0 0 28 17 0 0 44 45
,
 1 0 0 0 0 1 0 0 0 0 16 29 0 0 44 57
,
 66 66 0 0 7 59 0 0 0 0 44 45 0 0 16 28
,
 7 59 0 0 66 66 0 0 0 0 26 21 0 0 6 47
`G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,28,44,0,0,17,45],[1,0,0,0,0,1,0,0,0,0,16,44,0,0,29,57],[66,7,0,0,66,59,0,0,0,0,44,16,0,0,45,28],[7,66,0,0,59,66,0,0,0,0,26,6,0,0,21,47] >;`

Q8.D12 in GAP, Magma, Sage, TeX

`Q_8.D_{12}`
`% in TeX`

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

`G:=SmallGroup(192,949);`
`// by ID`

`G=gap.SmallGroup(192,949);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,672,85,36,451,1684,655,172,1013,404,285,124]);`
`// Polycyclic`

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

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