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## G = D4.D4order 64 = 26

### 1st non-split extension by D4 of D4 acting via D4/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — D4.D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×D4 — C4×D4 — D4.D4
 Lower central C1 — C2 — C2×C4 — D4.D4
 Upper central C1 — C22 — C42 — D4.D4
 Jennings C1 — C2 — C2 — C2×C4 — D4.D4

Generators and relations for D4.D4
G = < a,b,c,d | a4=b2=c4=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

Subgroups: 109 in 60 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Q8⋊C4, C4⋊C8, C4×D4, C4⋊Q8, C2×SD16, D4.D4
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C4⋊D4, C2×SD16, C8.C22, D4.D4

Character table of D4.D4

 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 -2 2 0 -2 2 0 0 0 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 orthogonal lifted from D4 ρ12 2 2 2 2 0 0 -2 -2 -2 -2 0 2 0 0 0 0 0 0 0 orthogonal lifted from D4 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 symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of D4.D4
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 4)(2 3)(5 8)(6 7)(9 12)(10 11)(13 16)(14 15)(17 19)(21 23)(25 27)(29 31)
(1 15 7 11)(2 16 8 12)(3 13 5 9)(4 14 6 10)(17 25 21 29)(18 26 22 30)(19 27 23 31)(20 28 24 32)
(1 31 3 29)(2 30 4 32)(5 25 7 27)(6 28 8 26)(9 17 11 19)(10 20 12 18)(13 21 15 23)(14 24 16 22)```

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

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

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

Matrix representation of D4.D4 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 0 1 0 0 16 0
,
 16 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
,
 13 0 0 0 0 4 0 0 0 0 1 0 0 0 0 1
,
 0 4 0 0 13 0 0 0 0 0 5 12 0 0 12 12
`G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[16,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[13,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[0,13,0,0,4,0,0,0,0,0,5,12,0,0,12,12] >;`

D4.D4 in GAP, Magma, Sage, TeX

`D_4.D_4`
`% in TeX`

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

`G:=SmallGroup(64,142);`
`// by ID`

`G=gap.SmallGroup(64,142);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,55,362,1444,376,88]);`
`// Polycyclic`

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

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