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## G = M4(2).5D4order 128 = 27

### 5th 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).5D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C2×C8⋊C22 — M4(2).5D4
 Lower central C1 — C2 — C22×C4 — M4(2).5D4
 Upper central C1 — C22 — C22×C4 — M4(2).5D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).5D4

Generators and relations for M4(2).5D4
G = < a,b,c,d | a8=b2=1, c4=d2=a4, bab=a5, cac-1=a3b, dad-1=ab, cbc-1=a4b, bd=db, dcd-1=c3 >

Subgroups: 424 in 181 conjugacy classes, 46 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×4], C4 [×3], C22 [×3], C22 [×13], C8 [×7], C2×C4 [×6], C2×C4 [×7], D4 [×10], Q8 [×8], C23, C23 [×7], C2×C8 [×2], C2×C8 [×5], M4(2) [×4], M4(2) [×4], D8 [×4], SD16 [×16], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×D4 [×6], C2×Q8, C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×4], C24, C22⋊C8 [×2], C4.D4 [×2], C4.10D4 [×2], C22×C8, C2×M4(2) [×3], C2×D8, C2×SD16 [×8], C2×Q16, C8⋊C22 [×4], C8.C22 [×4], C22×D4, C22×Q8, C2×C4○D4, C4.C42, (C22×C8)⋊C2, C2×C4.D4, C2×C4.10D4, C22×SD16, C2×C8⋊C22, C2×C8.C22, M4(2).5D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C232D4, D4.3D4 [×2], M4(2).5D4

Character table of M4(2).5D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J size 1 1 1 1 2 2 8 8 8 2 2 2 2 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 0 -2 2 2 -2 -2 0 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 -2 0 0 2 2 2 -2 -2 0 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 -2 2 -2 2 2 -2 0 2 -2 2 -2 0 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 0 -2 -2 2 2 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 0 -2 2 -2 2 0 0 0 0 0 0 0 2 0 0 0 0 -2 orthogonal lifted from D4 ρ14 2 -2 -2 2 -2 2 -2 2 0 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 2 -2 -2 0 0 0 -2 -2 2 2 0 0 0 -2 -2 2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ16 2 -2 -2 2 -2 2 0 0 0 -2 2 -2 2 0 0 0 0 0 0 0 -2 0 0 0 0 2 orthogonal lifted from D4 ρ17 2 2 2 2 -2 -2 0 0 -2 2 2 -2 -2 0 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 -2 -2 2 2 -2 0 0 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 -2 2 0 0 orthogonal lifted from D4 ρ19 2 -2 -2 2 2 -2 0 0 0 -2 2 2 -2 2 0 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 -2 -2 2 2 -2 0 0 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 2 -2 0 0 orthogonal lifted from D4 ρ21 2 2 2 2 2 2 0 0 0 -2 -2 -2 -2 0 0 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 -2 -2 -2 -2 0 0 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 complex lifted from D4.3D4 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 complex lifted from D4.3D4 ρ25 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 complex lifted from D4.3D4 ρ26 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 complex lifted from D4.3D4

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

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

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

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

Matrix representation of M4(2).5D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 16 0 0 0 0 16 0 0 0 1 0 0 0 0 0 0 16 0 0
,
 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 0 0 0 1
,
 4 0 0 0 0 0 13 13 0 0 0 0 0 0 0 0 5 5 0 0 0 0 5 12 0 0 5 12 0 0 0 0 12 12 0 0
,
 4 8 0 0 0 0 13 13 0 0 0 0 0 0 5 5 0 0 0 0 5 12 0 0 0 0 0 0 5 12 0 0 0 0 12 12

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,16,0,0,0,0,16,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,13,0,0,0,0,0,13,0,0,0,0,0,0,0,0,5,12,0,0,0,0,12,12,0,0,5,5,0,0,0,0,5,12,0,0],[4,13,0,0,0,0,8,13,0,0,0,0,0,0,5,5,0,0,0,0,5,12,0,0,0,0,0,0,5,12,0,0,0,0,12,12] >;`

M4(2).5D4 in GAP, Magma, Sage, TeX

`M_4(2)._5D_4`
`% in TeX`

`G:=Group("M4(2).5D4");`
`// GroupNames label`

`G:=SmallGroup(128,751);`
`// by ID`

`G=gap.SmallGroup(128,751);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,422,387,2019,1018,521,248,2804,718,172,4037,2028,1027,124]);`
`// Polycyclic`

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

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