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## G = M4(2).38D4order 128 = 27

### 2nd non-split extension by M4(2) of D4 acting via D4/C22=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — M4(2).38D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — Q8○M4(2) — M4(2).38D4
 Lower central C1 — C2 — C2×C4 — M4(2).38D4
 Upper central C1 — C2 — C22×C4 — M4(2).38D4
 Jennings C1 — C2 — C2 — C2×C4 — M4(2).38D4

Generators and relations for M4(2).38D4
G = < a,b,c,d | a8=b2=1, c4=d2=a4, bab=a5, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 396 in 223 conjugacy classes, 98 normal (36 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C4.D4, C4.10D4, C8.C4, C2×M4(2), C2×M4(2), C8○D4, C8○D4, C2×SD16, C2×SD16, C2×Q16, C2×Q16, C4○D8, C8⋊C22, C8⋊C22, C8.C22, C8.C22, C22×Q8, C2×C4○D4, C2×C4.10D4, M4(2).8C22, M4(2).C4, D4.3D4, D4.5D4, Q8○M4(2), C2×C8.C22, D8⋊C22, M4(2).38D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, C2×C4⋊D4, M4(2).38D4

Character table of M4(2).38D4

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L size 1 1 2 2 2 4 4 8 2 2 2 2 4 4 8 8 8 4 4 4 4 4 4 4 4 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 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 -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 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 -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 -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 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 -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 1 -1 -1 linear of order 2 ρ9 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 -1 1 1 linear of order 2 ρ10 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 1 1 -1 linear of order 2 ρ11 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 -1 -1 1 linear of order 2 ρ12 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 1 -1 -1 linear of order 2 ρ13 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 1 1 1 linear of order 2 ρ14 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 -1 1 -1 linear of order 2 ρ15 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 1 -1 1 linear of order 2 ρ16 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 -1 -1 -1 linear of order 2 ρ17 2 2 2 -2 -2 -2 -2 0 -2 2 -2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 -2 -2 2 2 0 -2 2 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 -2 2 0 0 0 -2 -2 2 2 0 0 0 0 0 0 2 -2 -2 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 -2 -2 2 2 -2 0 2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 2 2 -2 -2 0 0 0 2 -2 2 -2 0 0 0 0 0 0 -2 -2 2 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ22 2 2 2 -2 -2 0 0 0 2 -2 2 -2 0 0 0 0 0 0 2 2 -2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ23 2 2 -2 -2 2 -2 2 0 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ24 2 2 -2 -2 2 0 0 0 -2 -2 2 2 0 0 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ25 2 2 -2 2 -2 0 0 0 2 -2 -2 2 0 0 0 0 0 -2i 0 0 0 0 -2i 2i 2i 0 0 0 0 complex lifted from C4○D4 ρ26 2 2 2 2 2 0 0 0 -2 -2 -2 -2 0 0 0 0 0 2i 0 0 0 0 -2i -2i 2i 0 0 0 0 complex lifted from C4○D4 ρ27 2 2 2 2 2 0 0 0 -2 -2 -2 -2 0 0 0 0 0 -2i 0 0 0 0 2i 2i -2i 0 0 0 0 complex lifted from C4○D4 ρ28 2 2 -2 2 -2 0 0 0 2 -2 -2 2 0 0 0 0 0 2i 0 0 0 0 2i -2i -2i 0 0 0 0 complex lifted from C4○D4 ρ29 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of M4(2).38D4
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 13)(2 10)(3 15)(4 12)(5 9)(6 14)(7 11)(8 16)(17 29)(18 26)(19 31)(20 28)(21 25)(22 30)(23 27)(24 32)
(1 26 3 32 5 30 7 28)(2 25 4 31 6 29 8 27)(9 22 11 20 13 18 15 24)(10 21 12 19 14 17 16 23)
(1 12 5 16)(2 11 6 15)(3 10 7 14)(4 9 8 13)(17 30 21 26)(18 29 22 25)(19 28 23 32)(20 27 24 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), (1,13)(2,10)(3,15)(4,12)(5,9)(6,14)(7,11)(8,16)(17,29)(18,26)(19,31)(20,28)(21,25)(22,30)(23,27)(24,32), (1,26,3,32,5,30,7,28)(2,25,4,31,6,29,8,27)(9,22,11,20,13,18,15,24)(10,21,12,19,14,17,16,23), (1,12,5,16)(2,11,6,15)(3,10,7,14)(4,9,8,13)(17,30,21,26)(18,29,22,25)(19,28,23,32)(20,27,24,31)>;`

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

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

Matrix representation of M4(2).38D4 in GL8(𝔽17)

 2 15 2 2 0 0 0 0 2 2 15 2 0 0 0 0 15 15 15 2 0 0 0 0 2 15 15 15 0 0 0 0 0 0 0 0 2 15 2 2 0 0 0 0 2 2 15 2 0 0 0 0 15 15 15 2 0 0 0 0 2 15 15 15
,
 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 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 15 2 2 2 0 0 0 0 2 2 2 15 0 0 0 0 2 2 15 2 0 0 0 0 2 15 2 2 2 2 2 15 0 0 0 0 2 15 15 15 0 0 0 0 2 15 2 2 0 0 0 0 15 15 2 15 0 0 0 0
,
 2 2 2 15 0 0 0 0 2 15 15 15 0 0 0 0 2 15 2 2 0 0 0 0 15 15 2 15 0 0 0 0 0 0 0 0 15 2 2 2 0 0 0 0 2 2 2 15 0 0 0 0 2 2 15 2 0 0 0 0 2 15 2 2

`G:=sub<GL(8,GF(17))| [2,2,15,2,0,0,0,0,15,2,15,15,0,0,0,0,2,15,15,15,0,0,0,0,2,2,2,15,0,0,0,0,0,0,0,0,2,2,15,2,0,0,0,0,15,2,15,15,0,0,0,0,2,15,15,15,0,0,0,0,2,2,2,15],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,2,2,2,15,0,0,0,0,2,15,15,15,0,0,0,0,2,15,2,2,0,0,0,0,15,15,2,15,15,2,2,2,0,0,0,0,2,2,2,15,0,0,0,0,2,2,15,2,0,0,0,0,2,15,2,2,0,0,0,0],[2,2,2,15,0,0,0,0,2,15,15,15,0,0,0,0,2,15,2,2,0,0,0,0,15,15,2,15,0,0,0,0,0,0,0,0,15,2,2,2,0,0,0,0,2,2,2,15,0,0,0,0,2,2,15,2,0,0,0,0,2,15,2,2] >;`

M4(2).38D4 in GAP, Magma, Sage, TeX

`M_4(2)._{38}D_4`
`% in TeX`

`G:=Group("M4(2).38D4");`
`// GroupNames label`

`G:=SmallGroup(128,1801);`
`// by ID`

`G=gap.SmallGroup(128,1801);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,1018,2804,172,4037,1027,124]);`
`// Polycyclic`

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

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