Copied to
clipboard

## G = M4(2)⋊13D4order 128 = 27

### 7th 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)⋊13D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×M4(2) — M4(2)⋊13D4
 Lower central C1 — C2 — C2×C4 — M4(2)⋊13D4
 Upper central C1 — C2×C4 — C2×C42 — M4(2)⋊13D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2)⋊13D4

Generators and relations for M4(2)⋊13D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, ac=ca, dad=a-1b, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 348 in 168 conjugacy classes, 58 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C4 [×6], C4 [×9], C22 [×3], C22 [×8], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×17], D4 [×12], Q8 [×4], C23, C23 [×2], C42 [×4], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×2], M4(2) [×4], M4(2) [×2], C22×C4 [×3], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×8], C2.C42, C4×C8, C8⋊C4, C4≀C2 [×8], C2×C42, C2×C42 [×2], C2×C4⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4, C4⋊Q8, C2×M4(2) [×2], C2×C4○D4 [×2], C4×C4⋊C4, C4×M4(2), C2×C4≀C2 [×4], C22.26C24, M4(2)⋊13D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C4≀C2 [×2], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C24.3C22, C2×C4≀C2, C42⋊C22, M4(2)⋊13D4

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

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4J 4K ··· 4T 4U 4V 8A ··· 8H order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 4 4 8 ··· 8 size 1 1 1 1 2 2 8 8 1 1 1 1 2 ··· 2 4 ··· 4 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D4 D4 C4○D4 C4≀C2 C42⋊C22 kernel M4(2)⋊13D4 C4×C4⋊C4 C4×M4(2) C2×C4≀C2 C22.26C24 C4.4D4 C4⋊1D4 C4⋊Q8 C42 M4(2) C22×C4 C2×C4 C4 C2 # reps 1 1 1 4 1 4 2 2 2 4 2 4 8 2

Matrix representation of M4(2)⋊13D4 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 0 1 0 0 4 0
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 16
,
 16 16 0 0 2 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 15 16 0 0 0 0 0 13 0 0 4 0
`G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,4,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16],[16,2,0,0,16,1,0,0,0,0,1,0,0,0,0,1],[1,15,0,0,0,16,0,0,0,0,0,4,0,0,13,0] >;`

M4(2)⋊13D4 in GAP, Magma, Sage, TeX

`M_4(2)\rtimes_{13}D_4`
`% in TeX`

`G:=Group("M4(2):13D4");`
`// GroupNames label`

`G:=SmallGroup(128,712);`
`// by ID`

`G=gap.SmallGroup(128,712);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,100,1018,248,2028]);`
`// Polycyclic`

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

׿
×
𝔽