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## G = M4(2).31D4order 128 = 27

### 12nd non-split extension by M4(2) of 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).31D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C8○2M4(2) — M4(2).31D4
 Lower central C1 — C2 — C2×C4 — M4(2).31D4
 Upper central C1 — C22 — C22×C4 — M4(2).31D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).31D4

Generators and relations for M4(2).31D4
G = < a,b,c,d | a8=b2=1, c4=a4, d2=a6b, bab=a5, ac=ca, dad-1=a5b, bc=cb, dbd-1=a4b, dcd-1=c3 >

Subgroups: 332 in 148 conjugacy classes, 56 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×4], C22 [×3], C22 [×10], C8 [×4], C8 [×4], C2×C4 [×6], C2×C4 [×6], D4 [×6], Q8 [×6], C23, C23 [×6], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×4], C2×C8 [×3], M4(2) [×2], M4(2) [×5], SD16 [×8], C22×C4, C22×C4, C2×D4 [×2], C2×D4 [×5], C2×Q8 [×2], C2×Q8 [×5], C24, C4×C8, C8⋊C4, C4.D4 [×2], C4.10D4 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C8.C4 [×2], C42⋊C2, C22×C8, C2×M4(2) [×3], C2×SD16 [×4], C2×SD16 [×4], C22×D4, C22×Q8, C82M4(2), C2×C4.D4, C2×C4.10D4, C23.37D4, C23.38D4, C2×C8.C4, C22×SD16, M4(2).31D4
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], C4⋊D4 [×2], C4.4D4, C41D4, C24.3C22, D4.3D4 [×2], M4(2).31D4

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

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D 8E ··· 8J 8K 8L 8M 8N order 1 2 2 2 2 2 2 2 4 4 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 8 8 2 2 2 2 4 4 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 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D4 D4 C4○D4 C4○D4 D4.3D4 kernel M4(2).31D4 C8○2M4(2) C2×C4.D4 C2×C4.10D4 C23.37D4 C23.38D4 C2×C8.C4 C22×SD16 C2×SD16 C4⋊C4 C2×C8 M4(2) C2×C4 C23 C2 # reps 1 1 1 1 1 1 1 1 8 2 4 2 2 2 4

Matrix representation of M4(2).31D4 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 1 0 0 0 0 16 0 0 0
,
 1 0 0 0 0 0 0 1 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
,
 11 4 0 0 0 0 12 6 0 0 0 0 0 0 12 5 0 0 0 0 12 12 0 0 0 0 0 0 12 5 0 0 0 0 12 12
,
 6 13 0 0 0 0 13 11 0 0 0 0 0 0 0 0 5 5 0 0 0 0 5 12 0 0 12 5 0 0 0 0 5 5 0 0

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,16,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,12,0,0,0,0,4,6,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,5,12],[6,13,0,0,0,0,13,11,0,0,0,0,0,0,0,0,12,5,0,0,0,0,5,5,0,0,5,5,0,0,0,0,5,12,0,0] >;`

M4(2).31D4 in GAP, Magma, Sage, TeX

`M_4(2)._{31}D_4`
`% in TeX`

`G:=Group("M4(2).31D4");`
`// GroupNames label`

`G:=SmallGroup(128,709);`
`// by ID`

`G=gap.SmallGroup(128,709);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,100,2019,1018,2804,172,2028,124]);`
`// Polycyclic`

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

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