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## G = D8.A4order 192 = 26·3

### The non-split extension by D8 of A4 acting through Inn(D8)

Aliases: D8.A4, 2- 1+4⋊C6, SL2(𝔽3).13D4, Q8○D8⋊C3, C8.A44C2, C8○D42C6, C8.3(C2×A4), D4.A45C2, D4.3(C2×A4), C2.11(D4×A4), Q8.5(C3×D4), C4.6(C22×A4), C4.A4.17C22, C4○D4.3(C2×C6), SmallGroup(192,1019)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4○D4 — D8.A4
 Chief series C1 — C2 — Q8 — C4○D4 — C4.A4 — D4.A4 — D8.A4
 Lower central Q8 — C4○D4 — D8.A4
 Upper central C1 — C2 — C4 — D8

Generators and relations for D8.A4
G = < a,b,c,d,e | a8=b2=e3=1, c2=d2=a4, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a4c, ece-1=a4cd, ede-1=c >

Subgroups: 259 in 73 conjugacy classes, 19 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, C12, C2×C6, C2×C8, M4(2), D8, SD16, Q16, C2×Q8, C4○D4, C4○D4, C24, SL2(𝔽3), C3×D4, C8○D4, C2×Q16, C4○D8, C8.C22, 2- 1+4, C3×D8, C2×SL2(𝔽3), C4.A4, Q8○D8, C8.A4, D4.A4, D8.A4
Quotients: C1, C2, C3, C22, C6, D4, A4, C2×C6, C3×D4, C2×A4, C22×A4, D4×A4, D8.A4

Character table of D8.A4

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 8A 8B 8C 12A 12B 24A 24B 24C 24D size 1 1 4 4 6 4 4 2 6 12 12 4 4 16 16 16 16 2 2 12 8 8 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 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 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 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 linear of order 2 ρ5 1 1 -1 -1 1 ζ32 ζ3 1 1 -1 -1 ζ32 ζ3 ζ6 ζ65 ζ6 ζ65 1 1 1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 linear of order 6 ρ6 1 1 1 1 1 ζ3 ζ32 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 1 1 1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ7 1 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 1 1 1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ8 1 1 1 -1 1 ζ3 ζ32 1 1 1 -1 ζ3 ζ32 ζ65 ζ6 ζ3 ζ32 -1 -1 -1 ζ3 ζ32 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ9 1 1 1 -1 1 ζ32 ζ3 1 1 1 -1 ζ32 ζ3 ζ6 ζ65 ζ32 ζ3 -1 -1 -1 ζ32 ζ3 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ10 1 1 -1 1 1 ζ32 ζ3 1 1 -1 1 ζ32 ζ3 ζ32 ζ3 ζ6 ζ65 -1 -1 -1 ζ32 ζ3 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ11 1 1 -1 1 1 ζ3 ζ32 1 1 -1 1 ζ3 ζ32 ζ3 ζ32 ζ65 ζ6 -1 -1 -1 ζ3 ζ32 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ12 1 1 -1 -1 1 ζ3 ζ32 1 1 -1 -1 ζ3 ζ32 ζ65 ζ6 ζ65 ζ6 1 1 1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 linear of order 6 ρ13 2 2 0 0 -2 2 2 -2 2 0 0 2 2 0 0 0 0 0 0 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 0 0 -2 -1-√-3 -1+√-3 -2 2 0 0 -1-√-3 -1+√-3 0 0 0 0 0 0 0 1+√-3 1-√-3 0 0 0 0 complex lifted from C3×D4 ρ15 2 2 0 0 -2 -1+√-3 -1-√-3 -2 2 0 0 -1+√-3 -1-√-3 0 0 0 0 0 0 0 1-√-3 1+√-3 0 0 0 0 complex lifted from C3×D4 ρ16 3 3 -3 -3 -1 0 0 3 -1 1 1 0 0 0 0 0 0 3 3 -1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ17 3 3 3 3 -1 0 0 3 -1 -1 -1 0 0 0 0 0 0 3 3 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ18 3 3 -3 3 -1 0 0 3 -1 1 -1 0 0 0 0 0 0 -3 -3 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ19 3 3 3 -3 -1 0 0 3 -1 -1 1 0 0 0 0 0 0 -3 -3 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ20 4 -4 0 0 0 -2 -2 0 0 0 0 2 2 0 0 0 0 2√2 -2√2 0 0 0 -√2 √2 -√2 √2 symplectic faithful, Schur index 2 ρ21 4 -4 0 0 0 -2 -2 0 0 0 0 2 2 0 0 0 0 -2√2 2√2 0 0 0 √2 -√2 √2 -√2 symplectic faithful, Schur index 2 ρ22 4 -4 0 0 0 1-√-3 1+√-3 0 0 0 0 -1+√-3 -1-√-3 0 0 0 0 -2√2 2√2 0 0 0 -ζ83ζ3+ζ8ζ3 -ζ87ζ3+ζ85ζ3 -ζ83ζ32+ζ8ζ32 -ζ87ζ32+ζ85ζ32 complex faithful ρ23 4 -4 0 0 0 1-√-3 1+√-3 0 0 0 0 -1+√-3 -1-√-3 0 0 0 0 2√2 -2√2 0 0 0 -ζ87ζ3+ζ85ζ3 -ζ83ζ3+ζ8ζ3 -ζ87ζ32+ζ85ζ32 -ζ83ζ32+ζ8ζ32 complex faithful ρ24 4 -4 0 0 0 1+√-3 1-√-3 0 0 0 0 -1-√-3 -1+√-3 0 0 0 0 -2√2 2√2 0 0 0 -ζ83ζ32+ζ8ζ32 -ζ87ζ32+ζ85ζ32 -ζ83ζ3+ζ8ζ3 -ζ87ζ3+ζ85ζ3 complex faithful ρ25 4 -4 0 0 0 1+√-3 1-√-3 0 0 0 0 -1-√-3 -1+√-3 0 0 0 0 2√2 -2√2 0 0 0 -ζ87ζ32+ζ85ζ32 -ζ83ζ32+ζ8ζ32 -ζ87ζ3+ζ85ζ3 -ζ83ζ3+ζ8ζ3 complex faithful ρ26 6 6 0 0 2 0 0 -6 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4×A4

Smallest permutation representation of D8.A4
On 32 points
Generators in S32
```(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)
(2 8)(3 7)(4 6)(10 16)(11 15)(12 14)(17 19)(20 24)(21 23)(25 27)(28 32)(29 31)
(1 9 5 13)(2 10 6 14)(3 11 7 15)(4 12 8 16)(17 29 21 25)(18 30 22 26)(19 31 23 27)(20 32 24 28)
(1 30 5 26)(2 31 6 27)(3 32 7 28)(4 25 8 29)(9 18 13 22)(10 19 14 23)(11 20 15 24)(12 21 16 17)
(9 30 22)(10 31 23)(11 32 24)(12 25 17)(13 26 18)(14 27 19)(15 28 20)(16 29 21)```

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

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

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

Matrix representation of D8.A4 in GL4(𝔽7) generated by

 1 1 3 6 2 4 0 5 5 4 3 4 2 0 3 5
,
 6 0 0 0 6 6 6 3 5 6 4 5 1 2 1 5
,
 6 2 1 4 5 4 4 4 6 0 0 1 6 2 0 4
,
 6 6 3 5 2 1 6 0 0 0 5 3 0 0 3 2
,
 5 5 6 3 2 1 5 0 0 3 4 3 5 1 2 2
`G:=sub<GL(4,GF(7))| [1,2,5,2,1,4,4,0,3,0,3,3,6,5,4,5],[6,6,5,1,0,6,6,2,0,6,4,1,0,3,5,5],[6,5,6,6,2,4,0,2,1,4,0,0,4,4,1,4],[6,2,0,0,6,1,0,0,3,6,5,3,5,0,3,2],[5,2,0,5,5,1,3,1,6,5,4,2,3,0,3,2] >;`

D8.A4 in GAP, Magma, Sage, TeX

`D_8.A_4`
`% in TeX`

`G:=Group("D8.A4");`
`// GroupNames label`

`G:=SmallGroup(192,1019);`
`// by ID`

`G=gap.SmallGroup(192,1019);`
`# by ID`

`G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,197,3027,1522,248,438,172,775,285,124]);`
`// Polycyclic`

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

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