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## G = M4(2).7D4order 128 = 27

### 7th 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).7D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4⋊C4 — C23.67C23 — M4(2).7D4
 Lower central C1 — C2 — C22×C4 — M4(2).7D4
 Upper central C1 — C22 — C22×C4 — M4(2).7D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).7D4

Generators and relations for M4(2).7D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, cac-1=a3, dad=a-1b, cbc-1=dbd=a4b, dcd=a4c-1 >

Subgroups: 296 in 139 conjugacy classes, 42 normal (38 characteristic)
C1, C2 [×3], C2 [×3], C4 [×4], C4 [×8], C22 [×3], C22 [×5], C8 [×3], C2×C4 [×6], C2×C4 [×18], D4 [×4], Q8 [×6], C23, C23, C42 [×3], C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×6], C2×C8 [×2], M4(2) [×2], M4(2) [×3], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2×Q8 [×6], C4○D4 [×4], C2.C42 [×2], C4.10D4 [×2], D4⋊C4, Q8⋊C4, C4≀C2 [×2], C4.Q8, C2.D8, C2×C42, C2×C4⋊C4, C42⋊C2, C22⋊Q8 [×2], C22.D4 [×2], C4.4D4, C4⋊Q8, C2×M4(2) [×2], C22×Q8, C2×C4○D4, C426C4, C23.67C23, C2×C4.10D4, C23.36D4, C2×C4≀C2, M4(2)⋊C4, C23.38C23, M4(2).7D4
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, D4.8D4, D4.10D4, M4(2).7D4

Character table of M4(2).7D4

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 8A 8B 8C 8D size 1 1 1 1 2 2 8 2 2 2 2 4 4 4 4 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 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 2 2 -2 -2 -2 2 -2 -2 2 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 2 0 -2 -2 -2 -2 0 0 0 0 0 0 0 2 0 0 -2 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 -2 0 -2 2 2 -2 0 0 0 0 0 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 2 2 0 -2 -2 -2 -2 0 0 0 0 0 0 0 -2 0 0 2 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 -2 -2 0 -2 2 2 -2 0 0 0 0 0 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 -2 -2 2 2 -2 -2 2 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 -2 -2 2 2 -2 0 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 2 orthogonal lifted from D4 ρ16 2 -2 -2 2 2 -2 0 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 2 -2 orthogonal lifted from D4 ρ17 2 -2 -2 2 -2 2 0 -2 -2 2 2 0 0 0 0 0 0 0 0 2i -2i 0 0 0 0 0 complex lifted from C4○D4 ρ18 2 -2 -2 2 -2 2 0 2 2 -2 -2 -2i -2i 2i 2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ19 2 -2 -2 2 2 -2 0 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 -2i 2i 0 0 complex lifted from C4○D4 ρ20 2 -2 -2 2 -2 2 0 -2 -2 2 2 0 0 0 0 0 0 0 0 -2i 2i 0 0 0 0 0 complex lifted from C4○D4 ρ21 2 -2 -2 2 -2 2 0 2 2 -2 -2 2i 2i -2i -2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 -2 -2 2 2 -2 0 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 2i -2i 0 0 complex lifted from C4○D4 ρ23 4 4 -4 -4 0 0 0 0 0 0 0 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4.10D4, Schur index 2 ρ24 4 4 -4 -4 0 0 0 0 0 0 0 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4.10D4, Schur index 2 ρ25 4 -4 4 -4 0 0 0 0 0 0 0 2i -2i -2i 2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D4.8D4 ρ26 4 -4 4 -4 0 0 0 0 0 0 0 -2i 2i 2i -2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D4.8D4

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

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

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

Matrix representation of M4(2).7D4 in GL6(𝔽17)

 1 1 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 1 1 16 2 0 0 4 0 0 0 0 0 10 6 0 16
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 1 1 0 1
,
 13 13 0 0 0 0 0 4 0 0 0 0 0 0 1 1 16 2 0 0 0 0 1 0 0 0 0 16 0 0 0 0 16 16 0 16
,
 1 0 0 0 0 0 15 16 0 0 0 0 0 0 0 0 1 0 0 0 16 16 1 15 0 0 1 0 0 0 0 0 1 0 16 1

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,1,16,0,0,0,0,0,0,0,1,4,10,0,0,0,1,0,6,0,0,1,16,0,0,0,0,0,2,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,16,0,1,0,0,0,0,1,0,0,0,0,0,0,1],[13,0,0,0,0,0,13,4,0,0,0,0,0,0,1,0,0,16,0,0,1,0,16,16,0,0,16,1,0,0,0,0,2,0,0,16],[1,15,0,0,0,0,0,16,0,0,0,0,0,0,0,16,1,1,0,0,0,16,0,0,0,0,1,1,0,16,0,0,0,15,0,1] >;`

M4(2).7D4 in GAP, Magma, Sage, TeX

`M_4(2)._7D_4`
`% in TeX`

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

`G:=SmallGroup(128,770);`
`// by ID`

`G=gap.SmallGroup(128,770);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,456,422,387,58,2804,1411,718,172,4037]);`
`// 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^3,d*a*d=a^-1*b,c*b*c^-1=d*b*d=a^4*b,d*c*d=a^4*c^-1>;`
`// generators/relations`

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