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## G = C4⋊C4.96D4order 128 = 27

### 51st non-split extension by C4⋊C4 of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — C4⋊C4.96D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×M4(2) — C2×C4.D4 — C4⋊C4.96D4
 Lower central C1 — C2 — C22×C4 — C4⋊C4.96D4
 Upper central C1 — C22 — C22×C4 — C4⋊C4.96D4
 Jennings C1 — C2 — C2 — C22×C4 — C4⋊C4.96D4

Generators and relations for C4⋊C4.96D4
G = < a,b,c,d | a4=b4=d2=1, c4=a2, bab-1=dad=a-1, ac=ca, cbc-1=a2b-1, dbd=ab, dcd=a2c3 >

Subgroups: 408 in 148 conjugacy classes, 42 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×18], C8 [×5], C2×C4 [×2], C2×C4 [×4], C2×C4 [×2], D4 [×12], C23, C23 [×12], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×5], M4(2) [×6], D8 [×8], C22×C4, C2×D4 [×4], C2×D4 [×10], C24 [×2], C4.D4 [×4], D4⋊C4 [×4], C4⋊C8 [×2], C42⋊C2, C22×C8, C2×M4(2), C2×M4(2) [×2], C2×D8 [×6], C22×D4 [×2], C4.C42, C2×C4.D4 [×2], C23.37D4 [×2], C42.6C22, C22×D8, C4⋊C4.96D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×8], C23, C2×D4 [×4], C4○D4 [×3], C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C23.10D4, D4.4D4 [×2], C4⋊C4.96D4

Character table of C4⋊C4.96D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J size 1 1 1 1 2 2 8 8 8 8 2 2 2 2 8 8 4 4 4 4 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 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 ρ6 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 ρ7 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 ρ8 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 ρ9 2 -2 -2 2 -2 2 0 -2 0 2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 -2 2 2 -2 2 0 -2 0 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 -2 0 0 0 0 -2 2 2 -2 0 0 -2 2 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 -2 -2 0 0 0 0 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 -2 -2 2 -2 2 0 2 0 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 -2 -2 0 0 0 0 -2 2 2 -2 0 0 2 -2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 -2 -2 2 2 -2 -2 0 2 0 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ16 2 2 2 2 -2 -2 0 0 0 0 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ17 2 2 2 2 2 2 0 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 2i 0 0 -2i complex lifted from C4○D4 ρ18 2 2 2 2 2 2 0 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 -2i 0 0 2i complex lifted from C4○D4 ρ19 2 -2 -2 2 -2 2 0 0 0 0 2 -2 2 -2 0 0 0 0 0 0 0 2i 0 0 -2i 0 complex lifted from C4○D4 ρ20 2 -2 -2 2 -2 2 0 0 0 0 2 -2 2 -2 0 0 0 0 0 0 0 -2i 0 0 2i 0 complex lifted from C4○D4 ρ21 2 -2 -2 2 2 -2 0 0 0 0 -2 -2 2 2 0 0 0 0 0 0 -2i 0 0 2i 0 0 complex lifted from C4○D4 ρ22 2 -2 -2 2 2 -2 0 0 0 0 -2 -2 2 2 0 0 0 0 0 0 2i 0 0 -2i 0 0 complex lifted from C4○D4 ρ23 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 -2√2 0 2√2 0 0 0 0 0 0 0 orthogonal lifted from D4.4D4 ρ24 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 2√2 0 -2√2 0 0 0 0 0 0 orthogonal lifted from D4.4D4 ρ25 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 2√2 0 -2√2 0 0 0 0 0 0 0 orthogonal lifted from D4.4D4 ρ26 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√2 0 2√2 0 0 0 0 0 0 orthogonal lifted from D4.4D4

Smallest permutation representation of C4⋊C4.96D4
On 32 points
Generators in S32
```(1 12 5 16)(2 13 6 9)(3 14 7 10)(4 15 8 11)(17 27 21 31)(18 28 22 32)(19 29 23 25)(20 30 24 26)
(1 30 14 22)(2 19 15 27)(3 32 16 24)(4 21 9 29)(5 26 10 18)(6 23 11 31)(7 28 12 20)(8 17 13 25)
(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 9)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)```

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

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

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

Matrix representation of C4⋊C4.96D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 2 0 0 0 0 16 16 0 0 0 0 4 4 0 1 0 0 0 13 16 0
,
 13 9 0 0 0 0 0 4 0 0 0 0 0 0 10 0 6 11 0 0 12 0 11 0 0 0 3 3 12 5 0 0 0 3 5 12
,
 1 2 0 0 0 0 16 16 0 0 0 0 0 0 11 11 0 0 0 0 3 0 0 0 0 0 0 5 14 3 0 0 12 12 14 14
,
 1 0 0 0 0 0 16 16 0 0 0 0 0 0 11 11 0 0 0 0 3 6 0 0 0 0 0 5 14 3 0 0 12 12 3 3

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,4,0,0,0,2,16,4,13,0,0,0,0,0,16,0,0,0,0,1,0],[13,0,0,0,0,0,9,4,0,0,0,0,0,0,10,12,3,0,0,0,0,0,3,3,0,0,6,11,12,5,0,0,11,0,5,12],[1,16,0,0,0,0,2,16,0,0,0,0,0,0,11,3,0,12,0,0,11,0,5,12,0,0,0,0,14,14,0,0,0,0,3,14],[1,16,0,0,0,0,0,16,0,0,0,0,0,0,11,3,0,12,0,0,11,6,5,12,0,0,0,0,14,3,0,0,0,0,3,3] >;`

C4⋊C4.96D4 in GAP, Magma, Sage, TeX

`C_4\rtimes C_4._{96}D_4`
`% in TeX`

`G:=Group("C4:C4.96D4");`
`// GroupNames label`

`G:=SmallGroup(128,777);`
`// by ID`

`G=gap.SmallGroup(128,777);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,58,2019,1018,248,718,1027]);`
`// Polycyclic`

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

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