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## G = M4(2).41D4order 128 = 27

### 5th 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).41D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — Q8○M4(2) — M4(2).41D4
 Lower central C1 — C2 — C2×C4 — M4(2).41D4
 Upper central C1 — C4 — C22×C4 — M4(2).41D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).41D4

Generators and relations for M4(2).41D4
G = < a,b,c,d | a8=b2=c4=1, d2=a6, bab=a5, cac-1=dad-1=a-1b, cbc-1=dbd-1=a4b, dcd-1=a6c-1 >

Subgroups: 236 in 131 conjugacy classes, 54 normal (36 characteristic)
C1, C2, C2 [×5], C4 [×4], C4 [×6], C22 [×3], C22 [×4], C8 [×7], C2×C4 [×6], C2×C4 [×10], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C42 [×2], C42 [×2], C22⋊C4, C4⋊C4, C2×C8 [×2], C2×C8 [×7], M4(2) [×4], M4(2) [×10], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], C4≀C2 [×4], C4≀C2 [×2], C4⋊C8 [×2], C8.C4 [×2], C2×C42, C42⋊C2, C2×M4(2) [×4], C2×M4(2) [×2], C8○D4 [×4], C2×C4○D4, C426C4, M4(2)⋊4C4, C2×C4≀C2, C42⋊C22, C4⋊M4(2), M4(2).C4, Q8○M4(2), M4(2).41D4
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, M4(2).41D4

Permutation representations of M4(2).41D4
On 16 points - transitive group 16T215
Generators in S16
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 6)(4 8)(9 13)(11 15)
(1 4 5 8)(2 3 6 7)(9 12)(10 15)(11 14)(13 16)
(1 9 7 15 5 13 3 11)(2 16 8 14 6 12 4 10)```

`G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(9,13)(11,15), (1,4,5,8)(2,3,6,7)(9,12)(10,15)(11,14)(13,16), (1,9,7,15,5,13,3,11)(2,16,8,14,6,12,4,10)>;`

`G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(9,13)(11,15), (1,4,5,8)(2,3,6,7)(9,12)(10,15)(11,14)(13,16), (1,9,7,15,5,13,3,11)(2,16,8,14,6,12,4,10) );`

`G=PermutationGroup([(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,6),(4,8),(9,13),(11,15)], [(1,4,5,8),(2,3,6,7),(9,12),(10,15),(11,14),(13,16)], [(1,9,7,15,5,13,3,11),(2,16,8,14,6,12,4,10)])`

`G:=TransitiveGroup(16,215);`

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F ··· 4K 4L 4M 8A ··· 8H 8I 8J 8K 8L order 1 2 2 2 2 2 2 4 4 4 4 4 4 ··· 4 4 4 8 ··· 8 8 8 8 8 size 1 1 2 2 2 4 4 1 1 2 2 2 4 ··· 4 8 8 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 type + + + + + + + + + + - - + image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D4 Q8 Q8 D4 C4○D4 C4○D4 M4(2).41D4 kernel M4(2).41D4 C42⋊6C4 M4(2)⋊4C4 C2×C4≀C2 C42⋊C22 C4⋊M4(2) M4(2).C4 Q8○M4(2) C4≀C2 C2×C8 M4(2) C2×D4 C2×Q8 C4○D4 C2×C4 C23 C1 # reps 1 1 1 1 1 1 1 1 8 2 2 1 1 2 2 2 4

Matrix representation of M4(2).41D4 in GL4(𝔽5) generated by

 0 0 0 4 0 0 3 0 0 4 0 0 3 0 0 0
,
 2 0 0 3 0 2 4 0 0 3 3 0 4 0 0 3
,
 3 0 0 2 0 4 3 0 0 0 1 0 0 0 0 2
,
 0 4 0 0 2 0 0 0 3 0 0 1 0 3 3 0
`G:=sub<GL(4,GF(5))| [0,0,0,3,0,0,4,0,0,3,0,0,4,0,0,0],[2,0,0,4,0,2,3,0,0,4,3,0,3,0,0,3],[3,0,0,0,0,4,0,0,0,3,1,0,2,0,0,2],[0,2,3,0,4,0,0,3,0,0,0,3,0,0,1,0] >;`

M4(2).41D4 in GAP, Magma, Sage, TeX

`M_4(2)._{41}D_4`
`% in TeX`

`G:=Group("M4(2).41D4");`
`// GroupNames label`

`G:=SmallGroup(128,593);`
`// by ID`

`G=gap.SmallGroup(128,593);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,2019,1018,248,2804,718,172,124]);`
`// 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=d*a*d^-1=a^-1*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^6*c^-1>;`
`// generators/relations`

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