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## G = M4(2)⋊19D4order 128 = 27

### 6th semidirect product of M4(2) and D4 acting via D4/C22=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — M4(2)⋊19D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C4○D4 — Q8○M4(2) — M4(2)⋊19D4
 Lower central C1 — C2 — C2×C4 — M4(2)⋊19D4
 Upper central C1 — C4 — C22×C4 — M4(2)⋊19D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2)⋊19D4

Generators and relations for M4(2)⋊19D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, cac-1=a3b, dad=a-1b, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 340 in 166 conjugacy classes, 54 normal (26 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×13], D4 [×14], Q8 [×4], C23, C23 [×2], C23, C42 [×2], C42 [×3], C22⋊C4 [×6], C4⋊C4 [×4], C2×C8 [×6], M4(2) [×4], M4(2) [×6], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×D4 [×3], C2×Q8 [×2], C4○D4 [×8], C23⋊C4 [×2], C4≀C2 [×4], C2×C42, C42⋊C2 [×2], C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4, C4⋊Q8, C2×M4(2) [×2], C2×M4(2) [×2], C8○D4 [×4], C2×C4○D4 [×2], C426C4 [×2], C23.C23, C42⋊C22 [×2], C22.26C24, Q8○M4(2), M4(2)⋊19D4
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, M4(2)⋊19D4

Permutation representations of M4(2)⋊19D4
On 16 points - transitive group 16T251
Generators in S16
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 6)(4 8)(10 14)(12 16)
(2 4 6 8)(9 15 13 11)
(1 10)(2 9)(3 12)(4 11)(5 14)(6 13)(7 16)(8 15)```

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

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

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

`G:=TransitiveGroup(16,251);`

On 16 points - transitive group 16T299
Generators in S16
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 16)(2 13)(3 10)(4 15)(5 12)(6 9)(7 14)(8 11)
(1 14 5 10)(3 16 7 12)
(1 5)(2 15)(3 7)(4 9)(6 11)(8 13)```

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

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

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

`G:=TransitiveGroup(16,299);`

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F ··· 4K 4L ··· 4P 8A ··· 8H order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 2 2 2 4 4 8 1 1 2 2 2 4 ··· 4 8 ··· 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D4 D4 D4 C4○D4 M4(2)⋊19D4 kernel M4(2)⋊19D4 C42⋊6C4 C23.C23 C42⋊C22 C22.26C24 Q8○M4(2) C4.4D4 C4⋊1D4 C4⋊Q8 M4(2) C22×C4 C2×D4 C2×C4 C1 # reps 1 2 1 2 1 1 4 2 2 4 2 2 4 4

Matrix representation of M4(2)⋊19D4 in GL4(𝔽5) generated by

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

M4(2)⋊19D4 in GAP, Magma, Sage, TeX

`M_4(2)\rtimes_{19}D_4`
`% in TeX`

`G:=Group("M4(2):19D4");`
`// GroupNames label`

`G:=SmallGroup(128,616);`
`// by ID`

`G=gap.SmallGroup(128,616);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,1018,521,248,2804,1411,2028,1027]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^5,c*a*c^-1=a^3*b,d*a*d=a^-1*b,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;`
`// generators/relations`

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