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## G = M4(2).48D4order 128 = 27

### 12nd 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).48D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C8○D4 — M4(2).48D4
 Lower central C1 — C2 — C2×C4 — M4(2).48D4
 Upper central C1 — C22 — C22×C4 — M4(2).48D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).48D4

Generators and relations for M4(2).48D4
G = < a,b,c,d | a8=b2=c4=1, d2=a2b, bab=a5, cac-1=ab, dad-1=a5b, bc=cb, dbd-1=a4b, dcd-1=a2bc-1 >

Subgroups: 372 in 166 conjugacy classes, 56 normal (38 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×3], C22 [×3], C22 [×14], C8 [×7], C2×C4 [×6], C2×C4 [×7], D4 [×2], D4 [×11], Q8 [×2], Q8, C23, C23 [×7], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×9], M4(2) [×4], M4(2) [×8], D8 [×4], SD16 [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×D4 [×6], C2×Q8, C4○D4 [×4], C4○D4 [×2], C24, C4.D4 [×2], D4⋊C4 [×3], Q8⋊C4, C2×C4⋊C4, C22×C8, C22×C8, C2×M4(2) [×3], C2×M4(2), C8○D4 [×4], C2×D8, C2×SD16, C8⋊C22 [×4], C8⋊C22 [×2], C22×D4, C2×C4○D4, C4.C42, C22.C42, C2×C4.D4, C2×D4⋊C4, C23.36D4, C2×C8○D4, C2×C8⋊C22, M4(2).48D4
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], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, D4.3D4, D4.4D4, M4(2).48D4

Smallest permutation representation of M4(2).48D4
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 5)(3 7)(10 14)(12 16)(18 22)(20 24)(25 29)(27 31)
(1 16 29 22)(2 9 30 23)(3 14 31 20)(4 15 32 21)(5 12 25 18)(6 13 26 19)(7 10 27 24)(8 11 28 17)
(1 6 7 8 5 2 3 4)(9 18 11 24 13 22 15 20)(10 19 16 21 14 23 12 17)(25 30 31 32 29 26 27 28)```

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D4 D4 D4 D4 C4○D4 C4○D4 D4.3D4 D4.4D4 kernel M4(2).48D4 C4.C42 C22.C42 C2×C4.D4 C2×D4⋊C4 C23.36D4 C2×C8○D4 C2×C8⋊C22 C8⋊C22 C2×C8 M4(2) C2×D4 C2×Q8 C4○D4 C2×C4 C23 C2 C2 # reps 1 1 1 1 1 1 1 1 8 2 2 1 1 2 2 2 2 2

Matrix representation of M4(2).48D4 in GL6(𝔽17)

 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 0 6 11 0 0 0 0 3 0 0 0 11 6 0 0 0 0 14 0 0 0
,
 16 0 0 0 0 0 0 16 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
,
 0 13 0 0 0 0 13 0 0 0 0 0 0 0 0 6 0 0 0 0 3 0 0 0 0 0 0 0 11 6 0 0 0 0 14 6
,
 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 11 6 0 0 0 0 14 6 0 0 0 6 0 0 0 0 3 0 0 0

`G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,14,0,0,0,0,6,0,0,0,6,3,0,0,0,0,11,0,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,3,0,0,0,0,6,0,0,0,0,0,0,0,11,14,0,0,0,0,6,6],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,0,0,6,0,0,0,11,14,0,0,0,0,6,6,0,0] >;`

M4(2).48D4 in GAP, Magma, Sage, TeX

`M_4(2)._{48}D_4`
`% in TeX`

`G:=Group("M4(2).48D4");`
`// GroupNames label`

`G:=SmallGroup(128,639);`
`// by ID`

`G=gap.SmallGroup(128,639);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,2019,1018,521,1411,718,172,2028,1027,124]);`
`// Polycyclic`

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

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