Copied to
clipboard

## G = M4(2)⋊9D4order 128 = 27

### 3rd semidirect product of M4(2) and D4 acting via D4/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — M4(2)⋊9D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C4×M4(2) — M4(2)⋊9D4
 Lower central C1 — C2 — C2×C4 — M4(2)⋊9D4
 Upper central C1 — C22 — C2×C42 — M4(2)⋊9D4
 Jennings C1 — C2 — C2 — C2×C4 — M4(2)⋊9D4

Generators and relations for M4(2)⋊9D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=cac-1=a5, dad=a3, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 596 in 286 conjugacy classes, 108 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C4 [×8], C22, C22 [×2], C22 [×18], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×12], D4 [×18], Q8 [×10], C23, C23 [×10], C42 [×2], C42 [×2], C22⋊C4 [×12], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×8], D8 [×8], SD16 [×16], Q16 [×8], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4 [×4], C2×D4 [×9], C2×Q8 [×4], C2×Q8 [×5], C4○D4 [×8], C24, C4×C8 [×2], C8⋊C4 [×2], C2×C42, C2×C22⋊C4 [×2], C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×6], C4.4D4 [×2], C41D4, C4⋊Q8, C2×M4(2) [×2], C2×D8 [×4], C2×SD16 [×8], C2×Q16 [×4], C8⋊C22 [×8], C8.C22 [×8], C22×D4, C22×Q8, C2×C4○D4 [×2], C4×M4(2), C8.12D4 [×4], C83D4 [×2], C8.2D4 [×2], C2×C4.4D4, C22.26C24, C2×C8⋊C22 [×2], C2×C8.C22 [×2], M4(2)⋊9D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C41D4 [×4], C22×D4 [×3], C2×C41D4, D8⋊C22 [×2], M4(2)⋊9D4

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

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A ··· 4H 4I 4J 4K 4L 4M 4N 8A ··· 8H order 1 2 2 2 2 2 2 2 2 2 4 ··· 4 4 4 4 4 4 4 8 ··· 8 size 1 1 1 1 2 2 8 8 8 8 2 ··· 2 4 4 8 8 8 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D8⋊C22 kernel M4(2)⋊9D4 C4×M4(2) C8.12D4 C8⋊3D4 C8.2D4 C2×C4.4D4 C22.26C24 C2×C8⋊C22 C2×C8.C22 C42 M4(2) C22×C4 C2 # reps 1 1 4 2 2 1 1 2 2 2 8 2 4

Matrix representation of M4(2)⋊9D4 in GL6(𝔽17)

 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 0 4 0 0 0 13 13 13 9 0 0 0 13 0 0 0 0 0 4 0 4
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 16 16 0 16
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 13 0 0 0 13 13 0 13
,
 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 16 16 15 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,13,0,0,0,0,0,13,13,4,0,0,4,13,0,0,0,0,0,9,0,4],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,16,0,0,0,1,0,16,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,13,0,0,0,4,0,13,0,0,0,0,13,0,0,0,0,0,0,13],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,16,0,1,0,0,0,16,1,0,0,0,0,15,0,0,1] >;`

M4(2)⋊9D4 in GAP, Magma, Sage, TeX

`M_4(2)\rtimes_9D_4`
`% in TeX`

`G:=Group("M4(2):9D4");`
`// GroupNames label`

`G:=SmallGroup(128,1885);`
`// by ID`

`G=gap.SmallGroup(128,1885);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,520,1018,2804,172]);`
`// Polycyclic`

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

׿
×
𝔽