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## G = C24.28D4order 128 = 27

### 28th non-split extension by C24 of D4 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C24.28D4
 Chief series C1 — C2 — C22 — C23 — C24 — C22×D4 — C22.11C24 — C24.28D4
 Lower central C1 — C2 — C23 — C24.28D4
 Upper central C1 — C2 — C24 — C24.28D4
 Jennings C1 — C2 — C24 — C24.28D4

Generators and relations for C24.28D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=d, ab=ba, ac=ca, ad=da, eae-1=faf-1=acd, bc=cb, ebe-1=fbf-1=bd=db, ece-1=cd=dc, cf=fc, de=ed, df=fd, fef-1=bde-1 >

Subgroups: 492 in 190 conjugacy classes, 54 normal (20 characteristic)
C1, C2, C2 [×10], C4 [×11], C22 [×3], C22 [×4], C22 [×19], C2×C4 [×2], C2×C4 [×20], D4 [×14], C23 [×3], C23 [×6], C23 [×8], C42 [×2], C22⋊C4 [×6], C22⋊C4 [×13], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×7], C2×D4 [×4], C2×D4 [×12], C24 [×3], C23⋊C4 [×6], C2×C22⋊C4, C2×C22⋊C4 [×4], C2×C22⋊C4, C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C22≀C2, C4⋊D4 [×2], C22.D4 [×2], C22.D4, C22×D4 [×2], C23.9D4 [×2], C2×C23⋊C4, C2×C23⋊C4 [×2], C22.11C24, C233D4, C24.28D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, C24.28D4

Character table of C24.28D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 4Q size 1 1 2 2 2 2 2 2 2 4 4 8 4 4 4 4 4 4 4 4 4 4 8 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 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 -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 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 -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 -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 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 -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 1 1 -1 linear of order 2 ρ9 1 1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 -i i -i i i -i i -i 1 -1 -1 i i -i 1 -i linear of order 4 ρ10 1 1 -1 -1 1 -1 1 -1 1 -1 1 1 1 i i -i i -i i -i -i -1 -1 1 i -i -i -1 i linear of order 4 ρ11 1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 -1 -i i -i i i -i i -i 1 1 1 -i -i i -1 i linear of order 4 ρ12 1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 i i -i i -i i -i -i -1 1 -1 -i i i 1 -i linear of order 4 ρ13 1 1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 i -i i -i -i i -i i 1 -1 -1 -i -i i 1 i linear of order 4 ρ14 1 1 -1 -1 1 -1 1 -1 1 -1 1 1 1 -i -i i -i i -i i i -1 -1 1 -i i i -1 -i linear of order 4 ρ15 1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 -1 i -i i -i -i i -i i 1 1 1 i i -i -1 -i linear of order 4 ρ16 1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -i -i i -i i -i i i -1 1 -1 i -i -i 1 i linear of order 4 ρ17 2 2 -2 2 2 2 -2 -2 -2 2 -2 0 2 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 2 2 2 -2 -2 -2 -2 2 0 -2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 2 -2 -2 2 2 -2 0 0 0 0 0 2 -2 -2 0 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 -2 2 -2 -2 2 -2 2 2 0 -2 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 2 2 -2 2 -2 -2 2 -2 -2 -2 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ22 2 2 -2 2 -2 -2 2 2 -2 0 0 0 0 0 -2 2 2 0 0 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ23 2 2 -2 -2 -2 2 -2 2 2 0 0 0 0 -2 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ24 2 2 -2 -2 -2 2 -2 2 2 0 0 0 0 2 0 0 0 -2 -2 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ25 2 2 2 -2 -2 2 2 -2 -2 0 0 0 0 0 -2i -2i 2i 0 0 0 2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ26 2 2 2 -2 -2 2 2 -2 -2 0 0 0 0 0 2i 2i -2i 0 0 0 -2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ27 2 2 2 2 -2 -2 -2 -2 2 0 0 0 0 -2i 0 0 0 -2i 2i 2i 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ28 2 2 2 2 -2 -2 -2 -2 2 0 0 0 0 2i 0 0 0 2i -2i -2i 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ29 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C24.28D4
On 16 points - transitive group 16T218
Generators in S16
```(2 11)(3 13)(4 6)(5 12)(8 16)(9 14)
(2 16)(4 14)(6 9)(8 11)
(1 7)(2 11)(3 5)(4 9)(6 14)(8 16)(10 15)(12 13)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 14 15 4)(2 13 16 3)(5 11 12 8)(6 10 9 7)```

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

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

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

`G:=TransitiveGroup(16,218);`

On 16 points - transitive group 16T224
Generators in S16
```(1 5)(2 6)(3 10)(4 11)(7 13)(8 14)(9 16)(12 15)
(1 5)(2 9)(3 7)(4 11)(6 16)(8 14)(10 13)(12 15)
(1 15)(3 13)(5 12)(7 10)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 3 15 13)(2 9 16 6)(4 8 14 11)(5 10 12 7)```

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

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

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

`G:=TransitiveGroup(16,224);`

On 16 points - transitive group 16T268
Generators in S16
```(1 10)(2 8)(3 5)(4 9)(6 14)(7 15)(11 16)(12 13)
(1 3)(2 14)(4 16)(5 10)(6 8)(7 12)(9 11)(13 15)
(2 16)(4 14)(6 9)(8 11)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 3 15 13)(2 14 16 4)(5 10 12 7)(6 11 9 8)```

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

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

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

`G:=TransitiveGroup(16,268);`

On 16 points - transitive group 16T284
Generators in S16
```(2 16)(3 13)(5 12)(6 9)
(2 16)(4 14)(6 9)(8 11)
(2 16)(4 14)(5 12)(7 10)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 6 15 9)(2 5 16 12)(3 11 13 8)(4 10 14 7)```

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

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

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

`G:=TransitiveGroup(16,284);`

On 16 points - transitive group 16T295
Generators in S16
```(1 9)(2 10)(3 15)(4 14)(5 13)(6 16)(7 12)(8 11)
(1 8)(3 5)(9 11)(13 15)
(1 8)(4 6)(9 11)(14 16)
(1 8)(2 7)(3 5)(4 6)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 16 8 14)(2 13 7 15)(3 12 5 10)(4 9 6 11)```

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

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

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

`G:=TransitiveGroup(16,295);`

On 16 points - transitive group 16T304
Generators in S16
```(1 11)(2 12)(3 10)(4 9)(5 13)(6 14)(7 16)(8 15)
(1 5)(2 7)(3 6)(4 8)(9 15)(10 14)(11 13)(12 16)
(1 4)(5 8)(9 11)(13 15)
(1 4)(2 3)(5 8)(6 7)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 13 4 15)(2 12 3 10)(5 9 8 11)(6 16 7 14)```

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

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

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

`G:=TransitiveGroup(16,304);`

On 16 points - transitive group 16T318
Generators in S16
```(1 2)(3 9)(4 12)(5 8)(6 15)(7 14)(10 11)(13 16)
(1 3)(2 9)(4 11)(5 7)(6 16)(8 14)(10 12)(13 15)
(1 3)(2 9)(4 11)(5 15)(6 8)(7 13)(10 12)(14 16)
(1 10)(2 11)(3 12)(4 9)(5 13)(6 14)(7 15)(8 16)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 13 10 5)(2 6 11 14)(3 7 12 15)(4 16 9 8)```

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

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

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

`G:=TransitiveGroup(16,318);`

On 16 points - transitive group 16T326
Generators in S16
```(1 9)(2 16)(3 13)(4 12)(5 10)(6 15)(7 11)(8 14)
(2 8)(4 5)(10 12)(14 16)
(1 3)(2 5)(4 8)(6 7)(9 13)(10 16)(11 15)(12 14)
(1 7)(2 8)(3 6)(4 5)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 12 7 10)(2 11 8 9)(3 14 6 16)(4 13 5 15)```

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

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

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

`G:=TransitiveGroup(16,326);`

Matrix representation of C24.28D4 in GL8(ℤ)

 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1
,
 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1
,
 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1
,
 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0
,
 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0

`G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0] >;`

C24.28D4 in GAP, Magma, Sage, TeX

`C_2^4._{28}D_4`
`% in TeX`

`G:=Group("C2^4.28D4");`
`// GroupNames label`

`G:=SmallGroup(128,645);`
`// by ID`

`G=gap.SmallGroup(128,645);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,2019,1018,521,2804,1411,2028,1027]);`
`// Polycyclic`

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

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