Copied to
clipboard

## G = M4(2)⋊5D4order 128 = 27

### 5th semidirect product of M4(2) and 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 — M4(2)⋊5D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×D4 — C22.29C24 — M4(2)⋊5D4
 Lower central C1 — C2 — C22×C4 — M4(2)⋊5D4
 Upper central C1 — C2 — C22×C4 — M4(2)⋊5D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2)⋊5D4

Generators and relations for M4(2)⋊5D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, cac-1=ab, dad=a-1b, cbc-1=dbd=a4b, dcd=c-1 >

Subgroups: 488 in 190 conjugacy classes, 44 normal (36 characteristic)
C1, C2, C2 [×7], C4 [×4], C4 [×5], C22 [×3], C22 [×16], C8 [×4], C2×C4 [×6], C2×C4 [×9], D4 [×18], Q8 [×4], C23, C23 [×8], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×2], C2×C8 [×2], C2×C8, M4(2) [×2], M4(2) [×3], D8 [×6], SD16 [×8], Q16 [×2], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×D4 [×11], C2×Q8 [×2], C4○D4 [×8], C24, C23⋊C4 [×2], D4⋊C4 [×2], C4≀C2 [×2], C42⋊C2 [×2], C22≀C2 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C2×M4(2) [×2], C2×D8, C2×SD16, C4○D8 [×2], C8⋊C22 [×6], C8.C22 [×2], C22×D4, C2×C4○D4 [×2], M4(2)⋊4C4, C23.C23, C23.37D4, C42⋊C22, C22.29C24, C2×C8⋊C22, D8⋊C22, M4(2)⋊5D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C232D4, M4(2)⋊5D4

Character table of M4(2)⋊5D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D size 1 1 2 2 2 8 8 8 8 2 2 2 2 8 8 8 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 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 -2 2 -2 2 0 0 0 -2 -2 2 2 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 2 -2 0 0 2 -2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 2 0 0 0 0 -2 -2 -2 -2 2 0 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 -2 2 0 -2 0 0 2 -2 2 -2 0 0 0 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 -2 -2 0 0 0 0 2 -2 -2 2 0 0 0 0 0 0 2 0 -2 0 orthogonal lifted from D4 ρ14 2 2 -2 2 -2 0 0 -2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 -2 2 -2 -2 0 0 0 -2 -2 2 2 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ16 2 2 -2 -2 2 0 2 0 0 2 -2 2 -2 0 0 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ17 2 2 2 -2 -2 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 0 -2 0 2 orthogonal lifted from D4 ρ18 2 2 2 -2 -2 0 0 0 0 2 -2 -2 2 0 0 0 0 0 0 -2 0 2 0 orthogonal lifted from D4 ρ19 2 2 2 2 2 0 0 0 0 -2 -2 -2 -2 -2 0 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 -2 -2 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 0 2 0 -2 orthogonal lifted from D4 ρ21 2 2 -2 -2 2 0 0 0 0 -2 2 -2 2 0 0 0 0 2i -2i 0 0 0 0 complex lifted from C4○D4 ρ22 2 2 -2 -2 2 0 0 0 0 -2 2 -2 2 0 0 0 0 -2i 2i 0 0 0 0 complex lifted from C4○D4 ρ23 8 -8 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 M4(2)⋊5D4
On 16 points - transitive group 16T331
Generators in S16
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 5)(3 7)(10 14)(12 16)
(1 15)(2 16 6 12)(3 13)(4 14 8 10)(5 11)(7 9)
(1 15)(2 10)(3 9)(4 12)(5 11)(6 14)(7 13)(8 16)```

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

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

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

`G:=TransitiveGroup(16,331);`

On 16 points - transitive group 16T367
Generators in S16
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 13)(2 10)(3 15)(4 12)(5 9)(6 14)(7 11)(8 16)
(2 14 6 10)(3 7)(4 12 8 16)(9 13)
(1 11)(3 13)(5 15)(7 9)(10 14)(12 16)```

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

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

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

`G:=TransitiveGroup(16,367);`

Matrix representation of M4(2)⋊5D4 in GL8(ℤ)

 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 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
,
 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 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 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

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

M4(2)⋊5D4 in GAP, Magma, Sage, TeX

`M_4(2)\rtimes_5D_4`
`% in TeX`

`G:=Group("M4(2):5D4");`
`// GroupNames label`

`G:=SmallGroup(128,740);`
`// by ID`

`G=gap.SmallGroup(128,740);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,2019,1018,521,248,2804,718,172,4037]);`
`// 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*b,d*a*d=a^-1*b,c*b*c^-1=d*b*d=a^4*b,d*c*d=c^-1>;`
`// generators/relations`

Export

׿
×
𝔽