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## G = M4(2).32D4order 128 = 27

### 13rd 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).32D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C8○2M4(2) — M4(2).32D4
 Lower central C1 — C2 — C2×C4 — M4(2).32D4
 Upper central C1 — C22 — C22×C4 — M4(2).32D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).32D4

Generators and relations for M4(2).32D4
G = < a,b,c,d | a8=b2=1, c4=a4, d2=a2b, bab=a5, ac=ca, dad-1=a5b, bc=cb, dbd-1=a4b, dcd-1=a4c3 >

Subgroups: 412 in 158 conjugacy classes, 56 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C4 [×2], C22 [×3], C22 [×18], C8 [×4], C8 [×4], C2×C4 [×6], C2×C4 [×2], D4 [×12], C23, C23 [×12], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×4], C2×C8 [×3], M4(2) [×2], M4(2) [×5], D8 [×8], C22×C4, C2×D4 [×4], C2×D4 [×10], C24 [×2], C4×C8, C8⋊C4, C4.D4 [×4], D4⋊C4 [×4], C8.C4 [×2], C42⋊C2, C22×C8, C2×M4(2), C2×M4(2) [×2], C2×D8 [×4], C2×D8 [×4], C22×D4 [×2], C82M4(2), C2×C4.D4 [×2], C23.37D4 [×2], C2×C8.C4, C22×D8, M4(2).32D4
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, D4.4D4 [×2], M4(2).32D4

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

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

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

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

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 8 8 8 8 2 2 2 2 4 4 4 4 2 2 2 2 4 ··· 4 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 D4 D4 D4 C4○D4 C4○D4 D4.4D4 kernel M4(2).32D4 C8○2M4(2) C2×C4.D4 C23.37D4 C2×C8.C4 C22×D8 C2×D8 C4⋊C4 C2×C8 M4(2) C2×C4 C23 C2 # reps 1 1 2 2 1 1 8 2 4 2 2 2 4

Matrix representation of M4(2).32D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 6 3 1 15 0 0 0 1 0 0 0 0 1 5 13 14
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 6 3 0 16
,
 0 16 0 0 0 0 1 0 0 0 0 0 0 0 14 3 0 0 0 0 14 14 0 0 0 0 1 9 0 11 0 0 13 5 3 11
,
 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 6 3 1 15 0 0 0 16 0 0 0 0 1 4 7 14

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,6,0,1,0,0,0,3,1,5,0,0,1,1,0,13,0,0,0,15,0,14],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,6,0,0,0,1,0,3,0,0,0,0,16,0,0,0,0,0,0,16],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,14,14,1,13,0,0,3,14,9,5,0,0,0,0,0,3,0,0,0,0,11,11],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6,0,1,0,0,0,3,16,4,0,0,16,1,0,7,0,0,0,15,0,14] >;`

M4(2).32D4 in GAP, Magma, Sage, TeX

`M_4(2)._{32}D_4`
`% in TeX`

`G:=Group("M4(2).32D4");`
`// GroupNames label`

`G:=SmallGroup(128,710);`
`// by ID`

`G=gap.SmallGroup(128,710);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,436,2019,1018,2804,172,2028,124]);`
`// Polycyclic`

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

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