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## G = M4(2)⋊7D4order 128 = 27

### 1st semidirect product of M4(2) and D4 acting via D4/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — M4(2)⋊7D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C4×M4(2) — M4(2)⋊7D4
 Lower central C1 — C2 — C2×C4 — M4(2)⋊7D4
 Upper central C1 — C22 — C2×C42 — M4(2)⋊7D4
 Jennings C1 — C2 — C2 — C2×C4 — M4(2)⋊7D4

Generators and relations for M4(2)⋊7D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, ac=ca, dad=a-1, bc=cb, bd=db, dcd=c-1 >

Subgroups: 772 in 328 conjugacy classes, 112 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×8], C4 [×4], C22, C22 [×2], C22 [×28], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×10], D4 [×40], Q8 [×4], C23, C23 [×18], C42 [×2], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×8], D8 [×16], SD16 [×16], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×6], C2×D4 [×30], C2×Q8 [×2], C4○D4 [×8], C24 [×2], C4×C8 [×2], C8⋊C4 [×2], C2×C42, C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4, C41D4 [×4], C41D4 [×2], C4⋊Q8, C2×M4(2) [×2], C2×D8 [×8], C2×SD16 [×8], C8⋊C22 [×16], C22×D4 [×2], C22×D4 [×2], C2×C4○D4 [×2], C4×M4(2), C85D4 [×2], C84D4 [×2], C83D4 [×4], C2×C41D4, C22.26C24, C2×C8⋊C22 [×4], M4(2)⋊7D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C41D4 [×4], C8⋊C22 [×4], C22×D4 [×3], C2×C41D4, C2×C8⋊C22 [×2], M4(2)⋊7D4

Smallest permutation representation of M4(2)⋊7D4
On 32 points
Generators in S32
```(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)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)
(1 16 32 19)(2 9 25 20)(3 10 26 21)(4 11 27 22)(5 12 28 23)(6 13 29 24)(7 14 30 17)(8 15 31 18)
(1 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 15)(10 14)(11 13)(17 21)(18 20)(22 24)```

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 C8⋊C22 kernel M4(2)⋊7D4 C4×M4(2) C8⋊5D4 C8⋊4D4 C8⋊3D4 C2×C4⋊1D4 C22.26C24 C2×C8⋊C22 C42 M4(2) C22×C4 C4 # reps 1 1 2 2 4 1 1 4 2 8 2 4

Matrix representation of M4(2)⋊7D4 in GL6(ℤ)

 -1 -2 0 0 0 0 1 1 0 0 0 0 0 0 -1 1 1 0 0 0 0 0 1 -2 0 0 -1 0 0 2 0 0 -1 0 0 1
,
 -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 2 -2 -1 0 0 0 1 -1 0 -1
,
 1 2 0 0 0 0 -1 -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 1
,
 -1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 1 1 1 -2 0 0 1 0 0 -1

`G:=sub<GL(6,Integers())| [-1,1,0,0,0,0,-2,1,0,0,0,0,0,0,-1,0,-1,-1,0,0,1,0,0,0,0,0,1,1,0,0,0,0,0,-2,2,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,2,1,0,0,0,1,-2,-1,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,-1,0,0,0,0,2,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,1,0,0,0,-1,1,0,0,0,0,0,1,0,0,0,0,0,-2,-1] >;`

M4(2)⋊7D4 in GAP, Magma, Sage, TeX

`M_4(2)\rtimes_7D_4`
`% in TeX`

`G:=Group("M4(2):7D4");`
`// GroupNames label`

`G:=SmallGroup(128,1883);`
`// by ID`

`G=gap.SmallGroup(128,1883);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,184,1018,2804,172]);`
`// Polycyclic`

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

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