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## G = M4(2).42D4order 128 = 27

### 6th 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).42D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C8○D4 — M4(2).42D4
 Lower central C1 — C2 — C2×C4 — M4(2).42D4
 Upper central C1 — C2×C4 — C22×C8 — M4(2).42D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).42D4

Generators and relations for M4(2).42D4
G = < a,b,c,d | a8=b2=c4=1, d2=a6, bab=a5, cac-1=a3b, dad-1=a-1b, bc=cb, dbd-1=a4b, dcd-1=a6c-1 >

Subgroups: 236 in 135 conjugacy classes, 56 normal (46 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×6], C22 [×3], C22 [×6], C8 [×7], C2×C4 [×6], C2×C4 [×11], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C42 [×3], C22⋊C4, C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×10], M4(2) [×2], M4(2) [×7], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C4≀C2 [×2], C4⋊C8 [×2], C8.C4 [×2], C2×C42, C42⋊C2, C22×C8 [×2], C22×C8, C2×M4(2) [×2], C2×M4(2), C8○D4 [×4], C2×C4○D4, C22.7C42, C426C4, C23.24D4, C2×C4≀C2, C42.6C22, C2×C8.C4, C2×C8○D4, M4(2).42D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C23.8Q8, C8○D8, C8.26D4, M4(2).42D4

Smallest permutation representation of M4(2).42D4
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 23)(10 20)(11 17)(12 22)(13 19)(14 24)(15 21)(16 18)
(1 5)(2 26 6 30)(3 7)(4 28 8 32)(9 11 13 15)(10 20)(12 22)(14 24)(16 18)(17 19 21 23)(25 29)(27 31)
(1 19 7 17 5 23 3 21)(2 16 8 14 6 12 4 10)(9 25 15 31 13 29 11 27)(18 26 24 32 22 30 20 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), (1,27)(2,32)(3,29)(4,26)(5,31)(6,28)(7,25)(8,30)(9,23)(10,20)(11,17)(12,22)(13,19)(14,24)(15,21)(16,18), (1,5)(2,26,6,30)(3,7)(4,28,8,32)(9,11,13,15)(10,20)(12,22)(14,24)(16,18)(17,19,21,23)(25,29)(27,31), (1,19,7,17,5,23,3,21)(2,16,8,14,6,12,4,10)(9,25,15,31,13,29,11,27)(18,26,24,32,22,30,20,28)>;`

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G ··· 4L 4M 4N 8A 8B 8C 8D 8E ··· 8N 8O 8P 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 size 1 1 1 1 2 2 4 4 1 1 1 1 2 2 4 ··· 4 8 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 2 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 D4 D4 D4 Q8 C4○D4 C4○D4 C8○D8 C8.26D4 kernel M4(2).42D4 C22.7C42 C42⋊6C4 C23.24D4 C2×C4≀C2 C42.6C22 C2×C8.C4 C2×C8○D4 D4⋊C4 Q8⋊C4 C2×C8 M4(2) C4○D4 C4○D4 C2×C4 C23 C2 C2 # reps 1 1 1 1 1 1 1 1 4 4 2 2 2 2 2 2 8 2

Matrix representation of M4(2).42D4 in GL4(𝔽17) generated by

 0 16 0 0 16 0 0 0 0 0 0 8 0 0 9 0
,
 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 1
,
 1 0 0 0 0 16 0 0 0 0 16 0 0 0 0 4
,
 1 0 0 0 0 16 0 0 0 0 0 4 0 0 16 0
`G:=sub<GL(4,GF(17))| [0,16,0,0,16,0,0,0,0,0,0,9,0,0,8,0],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,4],[1,0,0,0,0,16,0,0,0,0,0,16,0,0,4,0] >;`

M4(2).42D4 in GAP, Magma, Sage, TeX

`M_4(2)._{42}D_4`
`% in TeX`

`G:=Group("M4(2).42D4");`
`// GroupNames label`

`G:=SmallGroup(128,598);`
`// by ID`

`G=gap.SmallGroup(128,598);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,2019,1018,521,248,2804,1411,718,172,124]);`
`// Polycyclic`

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

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