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## G = M4(2).12D4order 128 = 27

### 12nd 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).12D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C2×D4⋊C4 — M4(2).12D4
 Lower central C1 — C2 — C22×C4 — M4(2).12D4
 Upper central C1 — C22 — C22×C4 — M4(2).12D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).12D4

Generators and relations for M4(2).12D4
G = < a,b,c,d | a8=b2=c4=1, d2=a6b, bab=a5, cac-1=a-1, dad-1=a-1b, bc=cb, bd=db, dcd-1=a6bc-1 >

Subgroups: 280 in 114 conjugacy classes, 42 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×3], C22 [×3], C22 [×10], C8 [×6], C2×C4 [×6], C2×C4 [×4], D4 [×6], C23, C23 [×6], C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×5], M4(2) [×2], M4(2) [×5], C22×C4, C22×C4, C2×D4 [×2], C2×D4 [×5], C24, C4.D4 [×2], D4⋊C4 [×4], C4⋊C8 [×2], C4.Q8, C2.D8, C8.C4 [×2], C2×C4⋊C4, C42⋊C2, C22×C8, C2×M4(2) [×3], C22×D4, C22.C42, C2×C4.D4, C2×D4⋊C4, C23.37D4, C42.6C22, M4(2)⋊C4, C2×C8.C4, M4(2).12D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, C2×D4 [×3], C2×Q8, C4○D4 [×3], C4⋊D4 [×3], C22⋊Q8 [×3], C422C2, C23.Q8, D4.3D4, D4.4D4, M4(2).12D4

Character table of M4(2).12D4

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J size 1 1 1 1 2 2 8 8 2 2 2 2 8 8 8 8 4 4 4 4 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 0 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 -2 2 0 0 0 orthogonal lifted from D4 ρ10 2 -2 -2 2 2 -2 0 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 2 -2 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 -2 0 0 2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 -2 -2 0 0 -2 2 -2 2 0 0 0 0 2 -2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 -2 -2 0 0 -2 2 -2 2 0 0 0 0 -2 2 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 -2 -2 0 0 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 -2 -2 2 2 -2 -2 2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ16 2 -2 -2 2 2 -2 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ17 2 -2 -2 2 -2 2 0 0 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 -2i 0 2i complex lifted from C4○D4 ρ18 2 -2 -2 2 -2 2 0 0 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 2i 0 -2i complex lifted from C4○D4 ρ19 2 2 2 2 2 2 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 2i 0 0 0 -2i 0 complex lifted from C4○D4 ρ20 2 -2 -2 2 -2 2 0 0 2 2 -2 -2 0 0 -2i 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ21 2 -2 -2 2 -2 2 0 0 2 2 -2 -2 0 0 2i -2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 2 2 2 2 2 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 -2i 0 0 0 2i 0 complex lifted from C4○D4 ρ23 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 -2√2 0 2√2 0 0 0 0 0 0 0 orthogonal lifted from D4.4D4 ρ24 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 2√2 0 -2√2 0 0 0 0 0 0 0 orthogonal lifted from D4.4D4 ρ25 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√-2 0 2√-2 0 0 0 0 0 0 complex lifted from D4.3D4 ρ26 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 2√-2 0 -2√-2 0 0 0 0 0 0 complex lifted from D4.3D4

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

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

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

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

Matrix representation of M4(2).12D4 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 7 7 16 2 0 0 0 1 0 0 0 0 10 10 7 10
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 7 7 0 1
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 12 12 0 0 0 0 12 5 0 0 0 0 1 1 0 10 0 0 1 0 5 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 12 12 0 0 0 0 5 12 0 0 0 0 16 16 0 7 0 0 16 0 5 7

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,7,0,10,0,0,0,7,1,10,0,0,16,16,0,7,0,0,0,2,0,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,7,0,0,0,16,0,7,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,12,12,1,1,0,0,12,5,1,0,0,0,0,0,0,5,0,0,0,0,10,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,5,16,16,0,0,12,12,16,0,0,0,0,0,0,5,0,0,0,0,7,7] >;`

M4(2).12D4 in GAP, Magma, Sage, TeX

`M_4(2)._{12}D_4`
`% in TeX`

`G:=Group("M4(2).12D4");`
`// GroupNames label`

`G:=SmallGroup(128,795);`
`// by ID`

`G=gap.SmallGroup(128,795);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,168,141,64,422,387,2019,521,248,2804,4037,1027,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=a^6*b,b*a*b=a^5,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b,b*c=c*b,b*d=d*b,d*c*d^-1=a^6*b*c^-1>;`
`// generators/relations`

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