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G = M4(2).8D4order 128 = 27

8th non-split extension by M4(2) 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 — M4(2).8D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×M4(2) — M4(2).8C22 — M4(2).8D4
 Lower central C1 — C2 — C22×C4 — M4(2).8D4
 Upper central C1 — C2 — C22×C4 — M4(2).8D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).8D4

Generators and relations for M4(2).8D4
G = < a,b,c,d | a8=b2=d2=1, c4=a4, bab=a5, cac-1=a3, dad=ab, cbc-1=a4b, bd=db, dcd=a4c3 >

Subgroups: 344 in 137 conjugacy classes, 40 normal (34 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×2], C22 [×3], C22 [×11], C8 [×6], C2×C4 [×6], C2×C4 [×4], D4 [×10], Q8 [×2], C23, C23 [×7], C42, C22⋊C4, C4⋊C4, C2×C8 [×4], M4(2) [×4], M4(2) [×6], D8 [×4], SD16 [×4], C22×C4, C22×C4, C2×D4, C2×D4 [×2], C2×D4 [×6], C2×Q8, C4○D4 [×4], C24, C4.D4 [×3], C4.10D4, D4⋊C4 [×2], C4≀C2 [×2], C8.C4 [×2], C42⋊C2, C2×M4(2) [×4], C2×D8, C2×SD16, C8⋊C22 [×4], C22×D4, C2×C4○D4, M4(2)⋊4C4, C2×C4.D4, M4(2).8C22, C23.37D4, C42⋊C22, M4(2).C4, C2×C8⋊C22, M4(2).8D4
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, M4(2).8D4

Character table of M4(2).8D4

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 2 2 2 8 8 8 2 2 2 2 8 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 2 -2 2 -2 0 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 -2 -2 0 0 0 -2 2 2 -2 0 0 0 0 -2 0 0 0 2 0 0 orthogonal lifted from D4 ρ11 2 2 -2 2 -2 -2 0 0 2 -2 2 -2 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 -2 2 0 0 0 2 2 -2 -2 0 0 0 0 0 0 2 0 0 -2 0 orthogonal lifted from D4 ρ13 2 2 2 -2 -2 0 0 0 -2 2 2 -2 0 0 0 0 2 0 0 0 -2 0 0 orthogonal lifted from D4 ρ14 2 2 -2 -2 2 0 -2 2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 -2 -2 2 0 0 0 2 2 -2 -2 0 0 0 0 0 0 -2 0 0 2 0 orthogonal lifted from D4 ρ16 2 2 -2 -2 2 0 2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ17 2 2 2 2 2 0 0 0 -2 -2 -2 -2 0 0 0 -2i 0 2i 0 0 0 0 0 complex lifted from C4○D4 ρ18 2 2 -2 2 -2 0 0 0 -2 2 -2 2 0 0 0 0 0 0 0 2i 0 0 -2i complex lifted from C4○D4 ρ19 2 2 2 2 2 0 0 0 -2 -2 -2 -2 0 0 0 2i 0 -2i 0 0 0 0 0 complex lifted from C4○D4 ρ20 2 2 2 -2 -2 0 0 0 2 -2 -2 2 -2i 2i 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ21 2 2 -2 2 -2 0 0 0 -2 2 -2 2 0 0 0 0 0 0 0 -2i 0 0 2i complex lifted from C4○D4 ρ22 2 2 2 -2 -2 0 0 0 2 -2 -2 2 2i -2i 0 0 0 0 0 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).8D4
On 16 points - transitive group 16T411
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)
(1 12 3 10 5 16 7 14)(2 15 4 13 6 11 8 9)
(2 6)(3 7)(9 11)(10 16)(12 14)(13 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), (1,12,3,10,5,16,7,14)(2,15,4,13,6,11,8,9), (2,6)(3,7)(9,11)(10,16)(12,14)(13,15)>;`

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

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

`G:=TransitiveGroup(16,411);`

Matrix representation of M4(2).8D4 in GL8(ℤ)

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

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

M4(2).8D4 in GAP, Magma, Sage, TeX

`M_4(2)._8D_4`
`% in TeX`

`G:=Group("M4(2).8D4");`
`// GroupNames label`

`G:=SmallGroup(128,780);`
`// by ID`

`G=gap.SmallGroup(128,780);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,58,2019,1018,248,2804,172,1027]);`
`// Polycyclic`

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

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