Copied to
clipboard

## G = M4(2).40D4order 128 = 27

### 4th 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).40D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — Q8○M4(2) — M4(2).40D4
 Lower central C1 — C2 — C2×C4 — M4(2).40D4
 Upper central C1 — C4 — C22×C4 — M4(2).40D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).40D4

Generators and relations for M4(2).40D4
G = < a,b,c,d | a8=b2=1, c4=a4, d2=a6, bab=a5, cac-1=a3b, dad-1=a-1b, bc=cb, dbd-1=a4b, dcd-1=a2c3 >

Subgroups: 212 in 126 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×5], C8 [×10], C2×C4 [×2], C2×C4 [×6], C2×C4 [×4], D4 [×6], Q8 [×2], C23, C23 [×2], C2×C8 [×12], M4(2) [×8], M4(2) [×10], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C22⋊C8 [×2], C4.D4 [×2], C4.10D4 [×2], C8.C4 [×4], C22×C8, C2×M4(2), C2×M4(2) [×4], C2×M4(2) [×2], C8○D4 [×4], C2×C4○D4, C4.C42 [×2], (C22×C8)⋊C2, M4(2).8C22, M4(2).C4 [×2], Q8○M4(2), M4(2).40D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C23.8Q8, M4(2).40D4

Smallest permutation representation of M4(2).40D4
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 19)(2 24)(3 21)(4 18)(5 23)(6 20)(7 17)(8 22)(9 30)(10 27)(11 32)(12 29)(13 26)(14 31)(15 28)(16 25)
(1 28 7 26 5 32 3 30)(2 10 4 12 6 14 8 16)(9 19 15 17 13 23 11 21)(18 29 20 31 22 25 24 27)
(1 28 7 26 5 32 3 30)(2 10 8 16 6 14 4 12)(9 23 15 21 13 19 11 17)(18 25 24 31 22 29 20 27)```

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 8A ··· 8L 8M ··· 8R order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 8 ··· 8 8 ··· 8 size 1 1 2 2 2 4 4 1 1 2 2 2 4 4 4 ··· 4 8 ··· 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 D4 D4 Q8 C4○D4 M4(2).40D4 kernel M4(2).40D4 C4.C42 (C22×C8)⋊C2 M4(2).8C22 M4(2).C4 Q8○M4(2) C4.D4 C4.10D4 M4(2) C22×C4 C2×D4 C2×C4 C1 # reps 1 2 1 1 2 1 4 4 4 2 2 4 4

Matrix representation of M4(2).40D4 in GL4(𝔽17) generated by

 16 0 0 8 16 0 4 4 10 1 0 1 11 0 0 1
,
 13 8 0 0 13 4 0 0 16 1 0 4 0 1 13 0
,
 9 0 0 0 0 9 0 0 0 0 0 9 15 0 8 0
,
 9 0 0 0 9 8 0 0 0 0 0 9 0 0 9 0
`G:=sub<GL(4,GF(17))| [16,16,10,11,0,0,1,0,0,4,0,0,8,4,1,1],[13,13,16,0,8,4,1,1,0,0,0,13,0,0,4,0],[9,0,0,15,0,9,0,0,0,0,0,8,0,0,9,0],[9,9,0,0,0,8,0,0,0,0,0,9,0,0,9,0] >;`

M4(2).40D4 in GAP, Magma, Sage, TeX

`M_4(2)._{40}D_4`
`% in TeX`

`G:=Group("M4(2).40D4");`
`// GroupNames label`

`G:=SmallGroup(128,590);`
`// by ID`

`G=gap.SmallGroup(128,590);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,2019,1018,521,248,1411,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^4,d^2=a^6,b*a*b=a^5,c*a*c^-1=a^3*b,d*a*d^-1=a^-1*b,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=a^2*c^3>;`
`// generators/relations`

׿
×
𝔽