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## G = D4⋊2Q8order 64 = 26

### 2nd semidirect product of D4 and Q8 acting via Q8/C4=C2

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — D4⋊2Q8
 Chief series C1 — C2 — C4 — C2×C4 — C2×D4 — C4×D4 — D4⋊2Q8
 Lower central C1 — C2 — C2×C4 — D4⋊2Q8
 Upper central C1 — C22 — C42 — D4⋊2Q8
 Jennings C1 — C2 — C2 — C2×C4 — D4⋊2Q8

Generators and relations for D42Q8
G = < a,b,c,d | a4=b2=c4=1, d2=c2, bab=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=ab, dcd-1=c-1 >

Character table of D42Q8

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D size 1 1 1 1 4 4 2 2 2 2 4 4 4 8 8 4 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 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 linear of order 2 ρ7 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 linear of order 2 ρ9 2 2 2 2 0 0 2 -2 -2 2 0 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 -2 -2 -2 -2 0 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 -2 2 0 -2 2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ12 2 -2 2 -2 2 -2 0 -2 2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ13 2 -2 2 -2 0 0 0 2 -2 0 2i 0 -2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ14 2 -2 2 -2 0 0 0 2 -2 0 -2i 0 2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ15 2 -2 -2 2 0 0 -2 0 0 2 0 0 0 0 0 -√-2 √-2 -√-2 √-2 complex lifted from SD16 ρ16 2 -2 -2 2 0 0 2 0 0 -2 0 0 0 0 0 -√-2 -√-2 √-2 √-2 complex lifted from SD16 ρ17 2 -2 -2 2 0 0 2 0 0 -2 0 0 0 0 0 √-2 √-2 -√-2 -√-2 complex lifted from SD16 ρ18 2 -2 -2 2 0 0 -2 0 0 2 0 0 0 0 0 √-2 -√-2 √-2 -√-2 complex lifted from SD16 ρ19 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22

Smallest permutation representation of D42Q8
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)
(1 8)(2 7)(3 6)(4 5)(9 14)(10 13)(11 16)(12 15)(17 21)(18 24)(19 23)(20 22)(25 29)(26 32)(27 31)(28 30)
(1 16 5 10)(2 13 6 11)(3 14 7 12)(4 15 8 9)(17 26 23 32)(18 27 24 29)(19 28 21 30)(20 25 22 31)
(1 23 5 17)(2 22 6 20)(3 21 7 19)(4 24 8 18)(9 29 15 27)(10 32 16 26)(11 31 13 25)(12 30 14 28)```

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

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

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

Matrix representation of D42Q8 in GL4(𝔽17) generated by

 0 1 0 0 16 0 0 0 0 0 1 0 0 0 0 1
,
 0 16 0 0 16 0 0 0 0 0 16 0 0 0 0 16
,
 0 16 0 0 1 0 0 0 0 0 0 1 0 0 16 0
,
 12 5 0 0 5 5 0 0 0 0 0 4 0 0 4 0
`G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,16,0,0,16,0,0,0,0,0,16,0,0,0,0,16],[0,1,0,0,16,0,0,0,0,0,0,16,0,0,1,0],[12,5,0,0,5,5,0,0,0,0,0,4,0,0,4,0] >;`

D42Q8 in GAP, Magma, Sage, TeX

`D_4\rtimes_2Q_8`
`% in TeX`

`G:=Group("D4:2Q8");`
`// GroupNames label`

`G:=SmallGroup(64,157);`
`// by ID`

`G=gap.SmallGroup(64,157);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,240,121,55,362,1444,376,88]);`
`// Polycyclic`

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

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