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## G = M4(2).43D4order 128 = 27

### 7th 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).43D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C8○D4 — M4(2).43D4
 Lower central C1 — C2 — C2×C4 — M4(2).43D4
 Upper central C1 — C2×C4 — C22×C8 — M4(2).43D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).43D4

Generators and relations for M4(2).43D4
G = < a,b,c,d | a8=b2=1, c4=a4, d2=b, bab=a5, cac-1=a-1b, dad-1=a3b, cbc-1=a4b, bd=db, dcd-1=a4bc3 >

Subgroups: 308 in 162 conjugacy classes, 56 normal (46 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C22⋊C8, C4≀C2, C2×C42, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C22.7C42, C4.C42, (C22×C8)⋊C2, C2×C4≀C2, C2×C8○D4, C2×C4○D8, M4(2).43D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C23.23D4, C8○D8, C8.26D4, M4(2).43D4

Smallest permutation representation of M4(2).43D4
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)(9 13)(11 15)(18 22)(20 24)(26 30)(28 32)
(1 9 27 20 5 13 31 24)(2 16 28 19 6 12 32 23)(3 11 29 22 7 15 25 18)(4 10 30 21 8 14 26 17)
(2 4 6 8)(9 20 13 24)(10 19)(11 22 15 18)(12 21)(14 23)(16 17)(26 28 30 32)```

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G ··· 4L 4M 8A 8B 8C 8D 8E ··· 8N 8O 8P order 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 4 8 8 8 8 8 ··· 8 8 8 size 1 1 1 1 2 2 4 4 8 1 1 1 1 2 2 4 ··· 4 8 2 2 2 2 4 ··· 4 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 C4 C4 D4 D4 D4 C4○D4 C4○D4 C8○D8 C8.26D4 kernel M4(2).43D4 C22.7C42 C4.C42 (C22×C8)⋊C2 C2×C4≀C2 C2×C8○D4 C2×C4○D8 C2×D8 C2×SD16 C2×Q16 C2×C8 M4(2) C4○D4 C2×C4 C23 C2 C2 # reps 1 1 1 1 2 1 1 2 4 2 2 2 4 2 2 8 2

Matrix representation of M4(2).43D4 in GL4(𝔽17) generated by

 0 1 0 0 16 0 0 0 0 0 2 13 0 0 2 15
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 16
,
 16 0 0 0 0 1 0 0 0 0 13 8 0 0 6 4
,
 1 0 0 0 0 16 0 0 0 0 1 0 0 0 11 13
`G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,2,2,0,0,13,15],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,13,6,0,0,8,4],[1,0,0,0,0,16,0,0,0,0,1,11,0,0,0,13] >;`

M4(2).43D4 in GAP, Magma, Sage, TeX

`M_4(2)._{43}D_4`
`% in TeX`

`G:=Group("M4(2).43D4");`
`// GroupNames label`

`G:=SmallGroup(128,608);`
`// by ID`

`G=gap.SmallGroup(128,608);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,1018,521,248,1411,718,172,2028,1027]);`
`// Polycyclic`

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

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