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## G = M4(2).4D4order 128 = 27

### 4th 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).4D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C2×C8⋊C22 — M4(2).4D4
 Lower central C1 — C2 — C22×C4 — M4(2).4D4
 Upper central C1 — C22 — C22×C4 — M4(2).4D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).4D4

Generators and relations for M4(2).4D4
G = < a,b,c,d | a8=b2=d2=1, c4=a4, bab=a5, cac-1=a3b, dad=ab, cbc-1=a4b, bd=db, dcd=a4c3 >

Subgroups: 504 in 191 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C22⋊C8, C4.D4, C22×C8, C2×M4(2), C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C4.C42, (C22×C8)⋊C2, C2×C4.D4, C22×D8, C2×C8⋊C22, M4(2).4D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C22≀C2, C4⋊D4, C41D4, C232D4, D4.4D4, M4(2).4D4

Character table of M4(2).4D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 4A 4B 4C 4D 4E 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J size 1 1 1 1 2 2 8 8 8 8 8 2 2 2 2 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 2 0 0 0 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 -2 2 2 -2 0 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 0 2 -2 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 -2 0 0 0 0 0 2 2 -2 -2 0 2 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 -2 -2 2 2 -2 -2 0 0 0 2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 -2 -2 2 2 -2 0 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 0 -2 2 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 -2 -2 0 0 -2 0 0 -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 0 0 2 2 -2 -2 0 -2 -2 2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ16 2 -2 -2 2 -2 2 0 2 0 -2 0 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 2 0 0 0 0 0 -2 2 -2 2 0 0 0 0 0 -2 0 0 0 0 2 orthogonal lifted from D4 ρ18 2 -2 -2 2 -2 2 0 0 0 0 0 -2 2 -2 2 0 0 0 0 0 2 0 0 0 0 -2 orthogonal lifted from D4 ρ19 2 -2 -2 2 -2 2 0 -2 0 2 0 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 -2 -2 0 0 2 0 0 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 2 2 2 2 2 0 0 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 2i 0 0 -2i 0 complex lifted from C4○D4 ρ22 2 2 2 2 2 2 0 0 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 -2i 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 -2√2 0 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 0 0 2√2 -2√2 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 -2√2 2√2 0 0 0 0 0 0 0 0 orthogonal lifted from D4.4D4 ρ26 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√2 2√2 0 0 0 0 0 0 orthogonal lifted from D4.4D4

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

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

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

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

Matrix representation of M4(2).4D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 2 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 16 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 16 0 0 0 0 0 0 16
,
 13 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 6 11 0 0 0 0 3 11 0 0 0 11 0 0 0 0 14 0 0 0
,
 0 4 0 0 0 0 13 0 0 0 0 0 0 0 6 11 0 0 0 0 3 11 0 0 0 0 0 0 0 11 0 0 0 0 14 0

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,16,0,0,0,0,0,2,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,14,0,0,0,0,11,0,0,0,6,3,0,0,0,0,11,11,0,0],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,6,3,0,0,0,0,11,11,0,0,0,0,0,0,0,14,0,0,0,0,11,0] >;`

M4(2).4D4 in GAP, Magma, Sage, TeX

`M_4(2)._4D_4`
`% in TeX`

`G:=Group("M4(2).4D4");`
`// GroupNames label`

`G:=SmallGroup(128,750);`
`// by ID`

`G=gap.SmallGroup(128,750);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,2019,1018,521,248,2804,718,172,4037,2028,1027,124]);`
`// 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*b,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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