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## G = D4.9D4order 64 = 26

### 4th non-split extension by D4 of D4 acting via D4/C22=C2

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

Aliases: D4.9D4, Q8.9D4, C23.6D4, C423C22, M4(2)⋊2C22, 2+ 1+4.1C2, C4≀C23C2, C4.27(C2×D4), C4.4D42C2, C4.D42C2, C8.C221C2, (C2×C4).6C23, (C2×Q8)⋊2C22, C2.16C22≀C2, C4○D4.3C22, (C2×D4).8C22, C22.14(C2×D4), SmallGroup(64,136)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — D4.9D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×D4 — 2+ 1+4 — D4.9D4
 Lower central C1 — C2 — C2×C4 — D4.9D4
 Upper central C1 — C2 — C2×C4 — D4.9D4
 Jennings C1 — C2 — C2 — C2×C4 — D4.9D4

Generators and relations for D4.9D4
G = < a,b,c,d | a4=b2=c4=1, d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=a-1b, dbd-1=ab, dcd-1=a2c-1 >

Subgroups: 153 in 76 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×4], C22, C22 [×8], C8 [×2], C2×C4, C2×C4 [×6], D4 [×2], D4 [×8], Q8 [×2], Q8 [×2], C23 [×2], C23 [×2], C42, C22⋊C4 [×2], M4(2) [×2], SD16 [×2], Q16 [×2], C2×D4, C2×D4 [×4], C2×Q8, C4○D4 [×2], C4○D4 [×2], C4.D4, C4≀C2 [×2], C4.4D4, C8.C22 [×2], 2+ 1+4, D4.9D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, D4.9D4

Character table of D4.9D4

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 8A 8B size 1 1 2 4 4 4 4 2 2 4 4 4 4 8 8 8 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 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 linear of order 2 ρ7 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 linear of order 2 ρ9 2 2 -2 2 0 0 0 2 -2 0 -2 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 0 2 0 -2 -2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 0 -2 0 2 -2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 -2 0 0 0 2 -2 0 2 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 -2 0 0 2 0 -2 2 0 0 -2 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 -2 0 0 -2 0 -2 2 0 0 2 0 0 0 0 orthogonal lifted from D4 ρ15 4 -4 0 0 0 0 0 0 0 -2i 0 0 2i 0 0 0 complex faithful ρ16 4 -4 0 0 0 0 0 0 0 2i 0 0 -2i 0 0 0 complex faithful

Permutation representations of D4.9D4
On 16 points - transitive group 16T138
Generators in S16
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 12)(2 11)(3 10)(4 9)(5 14)(6 13)(7 16)(8 15)
(5 7)(6 8)(9 12 11 10)(13 14 15 16)
(1 5 3 7)(2 8 4 6)(9 16 11 14)(10 15 12 13)```

`G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,12)(2,11)(3,10)(4,9)(5,14)(6,13)(7,16)(8,15), (5,7)(6,8)(9,12,11,10)(13,14,15,16), (1,5,3,7)(2,8,4,6)(9,16,11,14)(10,15,12,13)>;`

`G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,12)(2,11)(3,10)(4,9)(5,14)(6,13)(7,16)(8,15), (5,7)(6,8)(9,12,11,10)(13,14,15,16), (1,5,3,7)(2,8,4,6)(9,16,11,14)(10,15,12,13) );`

`G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,12),(2,11),(3,10),(4,9),(5,14),(6,13),(7,16),(8,15)], [(5,7),(6,8),(9,12,11,10),(13,14,15,16)], [(1,5,3,7),(2,8,4,6),(9,16,11,14),(10,15,12,13)])`

`G:=TransitiveGroup(16,138);`

On 16 points - transitive group 16T145
Generators in S16
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 3)(5 6)(7 8)(9 11)(13 14)(15 16)
(1 6 12 13)(2 7 9 14)(3 8 10 15)(4 5 11 16)
(1 14 3 16)(2 13 4 15)(5 12 7 10)(6 11 8 9)```

`G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,3)(5,6)(7,8)(9,11)(13,14)(15,16), (1,6,12,13)(2,7,9,14)(3,8,10,15)(4,5,11,16), (1,14,3,16)(2,13,4,15)(5,12,7,10)(6,11,8,9)>;`

`G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,3)(5,6)(7,8)(9,11)(13,14)(15,16), (1,6,12,13)(2,7,9,14)(3,8,10,15)(4,5,11,16), (1,14,3,16)(2,13,4,15)(5,12,7,10)(6,11,8,9) );`

`G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,3),(5,6),(7,8),(9,11),(13,14),(15,16)], [(1,6,12,13),(2,7,9,14),(3,8,10,15),(4,5,11,16)], [(1,14,3,16),(2,13,4,15),(5,12,7,10),(6,11,8,9)])`

`G:=TransitiveGroup(16,145);`

On 16 points - transitive group 16T175
Generators in S16
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 8)(2 7)(3 6)(4 5)(9 13)(10 16)(11 15)(12 14)
(1 12 3 10)(2 9 4 11)(5 14)(6 15)(7 16)(8 13)
(1 11 3 9)(2 10 4 12)(5 13 7 15)(6 16 8 14)```

`G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,8)(2,7)(3,6)(4,5)(9,13)(10,16)(11,15)(12,14), (1,12,3,10)(2,9,4,11)(5,14)(6,15)(7,16)(8,13), (1,11,3,9)(2,10,4,12)(5,13,7,15)(6,16,8,14)>;`

`G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,8)(2,7)(3,6)(4,5)(9,13)(10,16)(11,15)(12,14), (1,12,3,10)(2,9,4,11)(5,14)(6,15)(7,16)(8,13), (1,11,3,9)(2,10,4,12)(5,13,7,15)(6,16,8,14) );`

`G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,8),(2,7),(3,6),(4,5),(9,13),(10,16),(11,15),(12,14)], [(1,12,3,10),(2,9,4,11),(5,14),(6,15),(7,16),(8,13)], [(1,11,3,9),(2,10,4,12),(5,13,7,15),(6,16,8,14)])`

`G:=TransitiveGroup(16,175);`

Matrix representation of D4.9D4 in GL4(𝔽5) generated by

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

D4.9D4 in GAP, Magma, Sage, TeX

`D_4._9D_4`
`% in TeX`

`G:=Group("D4.9D4");`
`// GroupNames label`

`G:=SmallGroup(64,136);`
`// by ID`

`G=gap.SmallGroup(64,136);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,362,963,489,255,117,730]);`
`// 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,c*b*c^-1=a^-1*b,d*b*d^-1=a*b,d*c*d^-1=a^2*c^-1>;`
`// generators/relations`

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